Курс математики для технических высших учебных заведений. Часть 2. Функции нескольких переменных. Интегральное исчисление. Теория поля: Учебное пособие [А. И. Мартыненко] (pdf) читать онлайн
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Лауреат второго Всероссийского конкурса НМС по математике
Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
В. А. ЛЯХОВСКИЙ, А. И. МАРТЫНЕНКО, В. Б. МИНОСЦЕВ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 2
Функции нескольких переменных.
Интегральное исчисление.
Теория поля
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ-ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Ляховский В. А., Мартыненко А. И., Миносцев В. Б.
К 93
Курс математики для технических высших учебных
заведений. Часть 2. Функции нескольких переменных.
Интегральное исчисление. Теория поля: Учебное пособие /
Под ред. В. Б. Миносцева, Е. А. Пушкаря. — 2-е изд.,
испр. — СПб.: Издательство «Лань», 2013. — 432 с.:
ил. — (Учебники для вузов. Специальная литература).
ISBN 9785811415595
Учебное пособие соответствует Государственному образовательному
стандарту. Пособие включает в себя лекции и практические занятия.
Вторая часть пособия содержит 25 лекций и 25 практических занятий
по следующим разделам: «Дифференциальное исчисление функций
нескольких переменных», «Интегральное исчисление функций одной
переменной», «Кратные интегралы», «Криволинейные интегралы и
теория поля».
Пособие предназначено для студентов технических, физикоматематических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è
ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è
íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ ïî
ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ; Ó. Ã. ÏÈÐÓÌÎÂ
ïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé ìàòåìàòèêè è ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò), ÷ëåí-êîððåñïîíäåíò ÐÀÍ,
çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
Охраняется законом РФ об авторском праве.
Воспроизведение всей книги или любой ее части запрещается без письменного
разрешения издателя.
Любые попытки нарушения закона
будут преследоваться в судебном порядке.
© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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b z− 2
2a
⎩
x = h.
⎨
2
= −y 2 ,
!" ! h # $
%& " Oyz & '
"$ b2
( $ % '
" % $ !
% &$ " $ '
Oyz " Oxz )'
% $ '
z = h h = 0
x2 y 2
− 2,
a2
b
z = h,
h=
x2
y2
−
= 1,
a2 h b2 h
z = h.
* + ! % $ " '
, h > 0 $ $ h < 0 %$ $ h = 0
z
y
0
x
Oxy
Oxy
x2
y2
−
= 0
a2 ⎧
b2
x
⎨ + y = 0,
a b
⎩ x − y = 0.
a b
Oxy
x y
+ = 0,
a b
z = 0,
x y
− = 0,
a b
z = 0,
Oxy
! "
#
⎧ x
⎨ +
a
⎩ x−
a
y
= kz,
b
y
1
= ,
b
k
⎧
⎨ x+
a
⎩ x−
a
y
1
= ,
by
l
= lz,
b
k l $
%
& '()
! "
y = −3x2 − 5z 2 #
x2
z2
+
,
y=−
1/3 1/5
! " #$ Oy % # &
' ' ( Oy
% # () ') y
1
1
' ( '
3
5
* ! "
x = 4z 2 − 16y 2 #
2
y2
z
−
,
1/4 1/16
+"
$, # Ox " $ # Oz " $
x=
$
2x2 + 3y 2 + 4z 2 − 5 = 0.
2x2 3y 2 4z 2
x2
y2
z2
+
+
=1⇔
+
+
= 1.
5
5
5
5/2 5/3 5/4
b=
5
c=
3
a =
5
2
5
4
3x2 − 4y 2 + 5z 2 − 6 = 0.
! "
#
3x2 2y 2 5z 2
x2
y2
z2
−
+
=1⇔
−
+
= 1.
6
3
6
6/3 3/2 6/5
#
# $
6
% # " Oy 3 65
y = 0
# " Oy
4y 2 + 4z 2 − 5x2 − 7 = 0.
! "
# &
4y 2 4z 2 5x2
y2
z2
x2
+
−
=1⇔
+
−
= 1.
7
7
7
7/4 7/4 7/5
# # ' $
% # " Ox 74 %
Ox
# ' " Ox
4y 2 + 5z 2 = 6x2 − 2.
6x2
z2
x2
5z 2
y2
− 3x2 = −1 ⇔
+
−
= −1.
2
1/2 2/5 1/3
2y 2 +
Ox
! " #$ Ox
%
3x2 + 3z 2 = 4y 2 − 4.
4y 2
3x2 3z 2
z2
x2
+
− y 2 = −1 ⇔
+
− y 2 = −1.
4
4
4/3 4/3
&
Oy
! " #$ Oy
'
3z 2 + 2y 2 − 5x = 0.
5x
%
x=
y2
z2
2y 2 3z 2
+
⇔x=
+
.
5
5
5/2 5/3
( )
Ox
! ) $ Ox
*
4y 2 − 3z 2 − 3x = 0.
Z
z
O1
Y
O
y
X
x
O1 XY
O1
x0 y0 z0 O1 (x0 ; y0 ; z0 )
!
" # $ %
& ' (%
x = X + x0 ;
y = Y + y0 ;
z = Z + z0 ,
)*"$
Y = y − y0 ;
Z = z − z0 .
)*!$
X = x − x0 ;
% +
& (& O Oxyz $
OXY Z $
,
' & x y z %
M % - X Y Z
. (
&
&
OX
% &
(% &
(%
cos ∠XOx = α11 ,
cos ∠XOy = α21 ,
cos ∠XOz = α31 .
L
+1
! " # $
! " #
L −1
! %&# ' (
Ox Oy ()
(
y
y
O
x
x
O
! !
" #$ % & '
! xy xz yz " a12 = a13 = a14 = 0$
( )
* %
* " +
3x2 + 2y 2 + z 2 − 6x + 4y − 4z + 5 = 0-
,
+ , - .&
(
-
3x2 + 2y 2 + z 2 − 6x + 4y − 4z + 5 = 0 ⇔
⇔ 3(x2 − 2x + 1) + 2(y 2 + 2y + 1) + (z 2 − 4z + 4) = 4 ⇔
⇔ 3(x−1)2 +2(y +1)2 +(z −2)2 = 4 ⇔
X = x − 1,
(x − 1)2 (y + 1)2 (z − 2)2
+
+
= 1.
4/3
4/2
4
Y = y + 1,
Z = z − 2.
2
X
Y 2 Z2
+
+
= 1.
4/3
2
4
√
2
a = √ , b = 2, c = 2.
3
X Y Z
x y z X = x − x0 Y = y − y0 Z = z − z0
! "#$!%!#& '
( '
P0 (x0; y0; z0)!
)
(
) P0 (1; −1; 2)
!
#*!+
4x2 + y 2 − 3z 2 + 16x + 2y + 6z + 6 = 0?
, -
'
(
4x2 + y 2 − 3z 2 + 16x + 2y + 6z + 6 = 0 ⇔
⇔ 4(x2 + 4x + 4) + (y 2 + 2y + 1) − 3(z 2 − 2z + 1) = 8 ⇔
⇔
(x + 2)2 (y + 1)2 (z − 1)2
+
−
= 1.
2
8
8/3
X = x + 2,
Y = y + 1,
Z = z − 1.
X2 Y 2
Z2
+
−
= 1.
2
8
8/3
OZ OZ
! "
# $ P0(−2; −1; 1)
Oz %
# !
& % '() *+ ,
!-- . a12 a13 a23
$
. $ - '() *+
! # . (, * /
0 % $ $
$ # 1
$
⎧
⎨ (a11 − λi )α1i + a12 α2i + a13 α3i = 0,
a12 α1i + (a22 − λi )α2i + a23 α3i = 0,
⎩
a13 α1i + a23 α2i + (a33 − λi )α3i = 0,
'(2 3+
'(2 (+
2
2
2
α1i
+ α2i
+ α3i
= 1,
$ # λ1 λ2 λ3 #
'(2 4+
a11 − λ
a12
a13
a12
a
−
λ
a23
22
a13
a23
a33 − λ
= 0.
'(2 4+
(2 5
⎛
⎞
a11 a12 a13
⎝ a12 a22 a23 ⎠ ,
a13 a23 a33
'(2 *6+
λi !"#$#%&' (
%)'*& # %)'+&
(X, Y, Z)
! " # $
%
a14 α11 +a24 α21 +a34 α31 =
$
= 0
& !
' ( & ! A $
) ( (det A = 0)
*
+
λ1 X 2 + λ2 Y 2 + λ3 Z 2 +
* + λ1 λ2 λ3 ,
&
⎛
det D
= 0,
det A
! )
⎞
a11 a12 a13
A = ⎝ a12 a22 a23 ⎠ ,
a13 a23 a33
det A , -
⎛
a11
⎜ a12
D=⎜
⎝ a13
a14
a12
a22
a23
a24
a13
a23
a33
a34
⎞
a14
a24 ⎟
⎟,
a34 ⎠
a44
!
det D , - & $
. &-
6x2 − 2y 2 + 6z 2 + 4zx + 8x − 4y − 8z + 1 = 0
⎛
6 0
⎝ 0 −2
2 0
⎞
2
0 ⎠
6
=0⇔
⇔ −(6 − λ)(2 + λ)(6 − λ) + 4(2 + λ) = 0 ⇔ (2 + λ) 4 − (6 − λ)2 = 0 ⇔
6−λ
0
2
0
−2
−
λ
0
2
0
6−λ
⇔ (2 + λ)(4 − 36 + 12λ − λ2 ) = 0 ⇔ (λ + 2)(λ − 8)(λ − 4) = 0 ⇔
⇔ λ1 = −2.
λ2 = 8,
λ3 = 4.
"
!
⎧
⎧
1
⎪
+
2α
=
0,
(6
−
4)α
α = −√ ,
⎪
⎪
11
31
⎪
⎪
⎨
⎨ 11
2
(−2 − 4)α21 = 0,
α21 = 0,
⇒
2α11 + (6 − 4)α31 = 0,
⎪
⎪
⎪
⎪
1
⎩ α2 + α2 + α2 = 1
⎪
⎩ α31 = √
11
21
31
2
⎧
⎧
1
⎪
(6 − 8)α12 + 2α32 = 0,
α =√ ,
⎪
⎪
⎪
⎪
⎨ 12
⎨
2
(−2 − 8)α22 = 0,
α22 = 0,
⇒
2α12 + (6 − 8)α32 = 0,
⎪
⎪
⎪
⎪
1
⎪
⎩ α2 + α2 + α2 = 1
⎩ α32 = √
12
22
32
2
⎧
⎧
(6
+
2)α
+
2α
=
0,
⎪
13
33
⎪
⎨
⎨ α13 = 0,
0 = 0,
α23 = 1,
⇒
2α13 + (6 + 2)α33 = 0,
⎪
⎩
⎪
α33 = 0.
⎩ α2 + α2 + α2 = 1
13
23
33
# L $ %
⎛
⎞
1
1
√ 0
−√
⎜
⎟
2
2
⎜
⎟
0 1 ⎟.
L=⎜ 0
⎝ 1
⎠
1
√
√ 0
2
2
' ( % X = LX !
⎧
1
1
⎪
x = − √ X + √ Y ,
⎪
⎪
⎨
2
2
y = Z
⎪
⎪
1
1
⎪
⎩ z = √ X + √ Y ,
2
2
&
4(X −
√
2)2 + 8Y
2
− 2(Z + 1)2 − 5 = 0.
⎧
√
⎨ X = X − 2,
Y = Y ,
⎩
Z = Z + 1,
4X 2 + 8Y 2 − 2Z 2 − 5 = 0 ⇔
2
2
2
Y
Z
X
+
−
= 1.
5/4 5/8 5/2
!
2x2 + 3y2 − 4x + 6y − 6z − 7 = 0
"#
$
%
2x2 − 4x + 3y 2 + 6y − 6z − 7 = 0 ⇔
⇔ 2(x2 − 2x + 1) + 3(y 2 + 2y + 1) − 6z − 12 = 0 ⇔
⇔z+2=
$
(x − 1)2 (y + 1)2
+
.
3
2
&
X = x − 1,
Y = y + 1,
' & &
2
Z=
(
2
2
Y
X
+
.
3
2
)
*
2
Z = z + 2.
2
2x + 5y + 2z − 2xy − 4zx + 2yz + 2x − 10y − 2z − 1 = 0,
⎛
⎞
2 −1 −2
1 ⎠
A = ⎝ −1 5
−2 1
2
2 − λ −1
−2
−1 5 − λ
1
−2
1
2−λ
=0
! "! # $ % !% &% ! '
!% # !# ! !'
((
−λ −1
−2
0 5−λ
1
−λ
1
2−λ
−2λ
0
λ
=0⇔ 0
5
−
λ
1
−λ
1
2−λ
⇔
−λ(λ2 − 9λ + 18) = 0 ⇔ λ(λ − 3)(λ − 6) = 0
λ1 = 6,
)
λ2 = 3,
λ3 = 0.
⎧
1
⎪
⎧
⎪
α11 = − √ ,
⎪
(2
−
3)α
−
1α
−
2α
=
0,
⎪
⎪
11
21
31
6
⎪
⎪
⎨
⎨
2
−1α11 + (5 − 3)α21 + 1α31 = 0,
α21 = √ ,
⇒
−2α11 + 1α21 + (2 − 3)α31 = 0,
⎪
⎪
6
⎪
⎪
⎪
⎩ α2 + α2 + α2 = 1
1
⎪
⎪
11
21
31
⎩ α31 = √ .
6
⎧
1
⎧
⎪
⎪
α12 = − √ ,
⎪
(2
−
6)α
−
1α
−
2α
=
0,
⎪
⎪
12
22
32
3
⎪
⎪
⎨
⎨
1
−1α12 + (5 − 6)α22 + 1α32 = 0,
α22 = − √ ,
⇒
⎪
⎪
12 + 1α22 + (2 − 6)α32 = 0,
3
⎪ −2α
⎪
⎩
⎪
2
2
2
1
⎪
+ α22
+ α32
=1
α12
⎪
⎩ α32 = √
3
⎧
⎧
1
⎪
2α13 − 1α23 − 2α33 = 0,
α =√ ,
⎪
⎪
⎪
⎪
⎨ 13
⎨
2
−1α13 + 5α23 + 1α33 = 0,
α23 = 0,
⇒
−2α13 + 1α23 + 2α33 = 0,
⎪
⎪
⎪
⎪
⎪ α = √1 .
⎩ α2 + α2 + α2 = 1
⎩
33
13
23
33
2
! " # $% L
&
⎛
⎞
1
1
−√ −√
⎜
6
3
⎜ 2
1
⎜ √
√
−
L=⎜
⎜
6
3
⎝ 1
1
√
√
6
3
1
√
2
0
⎟
⎟
⎟
⎟.
⎟
1 ⎠
√
2
' $% ! # X = LX
⎧
1
1
1
⎪
⎪
x = − √ X − √ Y + √ Z ,
⎪
⎪
6
3
2
⎪
⎨
2
1
y = √ X − √ Y ,
⎪
6
3
⎪
⎪
1
1
1
⎪
⎪
⎩ z = √ X + √ Y + √ Z ,
6
3
2
% ( ) #
! " &
! !
2X 2 + Y 2 − 2 = 0 ⇔ X 2 +
Y2
= 1.
2
x2 − 2y 2 + z 2 + 4xy − 8zx − 4yz − 14x − 4y + 14z + 16 = 0,
& *" +, ! # -
# &
⎛
⎞
1
2 −4
A = ⎝ 2 −2 −2 ⎠ .
−4 −2 1
! !
1−λ
2
−4
2
−2
−
λ
−2
−4
−2
1−λ
= 0,
λ1 = −3,
λ3 = −3.
λ2 = 6,
! "
# $ % & #
L
⎛
⎞
1
2
4
√
−√ −
⎜
3
3 5 ⎟
⎜ 25
1
2 ⎟
⎜ √
√ ⎟
−
L=⎜
⎟.
3 3√ 5 ⎟
⎜
5
⎝
5 ⎠
2
0
3
3
' # L
⎧
X
2
4
⎪
⎪
x = − √ − Y + √ Z ,
⎪
⎪
3
5
3 5
⎪
⎨
2
2
2
y = √ X + Y + √ Z ,
⎪
3
5
3
5
⎪
√
⎪
⎪
⎪
⎩ z = 2 Y + 5 Z ,
3
3
# $ X ! Y ! Z
# ! ("
# X Y ! X Z ! Y Z )
* #
& $ ! # $ $
$
2
Y
+ Z 2 = 0.
1/2
+ ! * det A = 54, det D = 0
X 2 − 2Y 2 + Z 2 = 0 ⇔ X 2 −
) # #
$ $
# &
, x2 + 2x + 2z2 − 4z + 4y2 + 2 = 0.
- x2 − 2x + y2 + 2y + z2 − 1 = 0.
. x2 + 4z2 − 2y2 − 4 = 0.
3y2 − 6y − x2 + 3z2 = 0.
3y2 − 6z2 − 2x2 − 6 = 0.
/ z2 + 2z − 3x2 + 6x − 3y2 − 5 = 0.
01 3x2 + 2z2 − 6y = 0.
x2 + 9y 2 + 18y + 9z + 9 = 0.
3x2 − 2z 2 − 6y = 0.
x2 +y 2 +5z 2 −6xy +2zx−2yz −4x+8y −12z +14 = 0
4x2 + 5y 2 + 6z 2 − 4xy + 4yz + 4x + 6y + 4z − 27 = 0.
x2 + 2y 2 + 3z 2 + 2x − 4y − 12z + 9 = 0.
y 2 + 2y + z 2 − 2z − 4x + 2 = 0.
! "! # $"
% $" & % $" $
' &
$$"
! ! ( $") *
$ + # &" δ $"$" " x0 , " $
" " x0 (x0 − δ; x0 + δ) - &"
&
. δ
P0 (x0 ; y0 )
δ(P0 ) = P (x; y)| (x − x0 )2 + (y − y0 )2 < δ
δ
/0# & " P 1"! δ $"$" "$& " " P0 $
$"& )' δ
P0 (x0 ; y0 )
!
ε
!
#
.
b
P (x; y)
P0 !
|f (x; y) − b| < ε
!
"
lim f (x; y) = b
x→x0
y→y0
z = f (x; y) P → P0
δ
lim f (P ) = b.
P →P0
Plim
→P
0
∃(ε > 0) ∀(δ(P0 ))∃(P ∈ δ(P0 ),
f (P ) = b
P = P0 ) ⇒ |f (P ) − b| < ε.
! "! #
! !
$ % & !' ! (
! % & !' !
!$( % %
) *+, *, !
lim f (x; y) = b ⇔ δ→0
lim f (x; y) = b δ = (x − x0 )2 + (y − y0 )2 %! P P0 ./ !
"! # !& & !
*,
b
x→x0
y→y0
z = f (x; y) P → P0
P0
P0 δ→0
lim f (x; y) = b
δ=
(x − x0 )2 + (y − y0 )2
+ ! lim
x→0
y→0
x2 + y 2
x2 + y 2 + 4 − 2
0
1 2 x0 = 0
⇒δ=
x2 + y 2
lim
x→0
y→0
= lim
δ→0
δ2
δ2
=
= lim √
x2 + y 2 + 4 − 2 δ→0 δ 2 + 4 − 2
x2 + y 2
√
√
δ2 + 4 + 2
2 + 4 + 2 = 4.
=
lim
δ
δ→0
δ2 + 4 − 4
x2 + y 2
x2 + y 2 + 4 − 2
P → P0
P0 (0; 0)
y0 = 0 P0 (0; 0) ⇒
x→x
lim f (x; y)
0
y→y0
lim
x→x0
lim f (x; y)
y→y0
lim
y→y0
lim f (x; y) ,
x→x0
!
" #
$% & x→x
lim f (x; y) ∀y ∈ δ
0
y0,
y→y0
y = y0 ∃ lim f (x; y)
∀x ∈ δ x0,
x→x
∃ lim f (x; y) ∃ lim lim f (x; y) ∃ lim lim f (x; y)
y→y
y→y x→x
x→x y→y
lim lim f (x; y) = lim lim f (x; y) = lim f (x; y)
y→y x→x
x→x y→y
x = x0
$% '
0
0
0
0
0
0
0
0
0
x→x0
y→y0
0
( " )
lim lim
x2 + y 2
= lim
=
x2 + y 2 + 4 − 2 y→0 y 2 + 4 − 2
y2
y2 + 4 + 2
2 + 4 + 2 = 4.
=
lim
y
= lim
y→0
y→0
y2 + 4 − 4
x2 + y 2
x→0
lim lim
= 4
2
y→0
x + y2 + 4 − 2
y→0 x→0
*
y2
+
" , - ./
x→x
lim lim
x→x0 +0
0 −0
! lim f (x) #
x→x0
0 2x lim
f (x; y) !
x→0
y = f (x)
y→0
P (x; y) P0 (x0; y0 )
2
2
y
$% $ ! lim xx2 −
+ y2
x→0
y→0
y = λx
x2 − y 2
x2 − λ2 x2
y = λx lim 2 2 = x→0
lim 2
=
2 2
x→0
y→0
2
=
x +y
x +λ x
1−λ
1 + λ2
λ
!" #
!" $ f (x; y)
P → P0
%&
' ( &
)
lim f (P ) = b ⇔ lim (f (P ) − b) = 0 ⇔ f (P ) − b
P →P0
P →P0
P → P0
* )
+, ! )
!" - n u = f (P )
P0
Plim
f (P ) = f (P0 )
→P
!" . P0 u = f (P )
0
/ ) &
& & ) +
&
0 +, +
' )
' +,
1
!" 2 ! n f1(P ) f2(P )
P0
f1 (P ) + f2 (P )
f1(P )/f2(P )
"
# f1(P ) − f2(P ) f1(P ) · f2(P )
f2(P0) = 0
3 4
')
, +,
'
P0
f (P )
z = x − 1y + 1
x − y + 1 = 0
! y = x + 1 ! " ! # $
% &' ( ! ! ) *
+ * #* y = x + 1
, - # ! %
! $
! ! * . " ! #$
%
) / / " !
* ! #
z = f (P )
! D
" ! f (P )
P0 # P ! P → P0
! D
$ z = f (P )
! D %
012 % ∃ N > 0 : |f (P )| N ∀P ∈ D&
032
! '( m '( M
% ∃ P1 ∈ D : f (P1) = m ∃ P2 ∈ D : f (P2) = M &
0 2 # ! m M !
)
% ∀ c ∈ [m; M] ∃ P0 : f (P0) = c
4 z = 1 − x2 − y2 2 2
! D = {(x; y)|x + y 1} *
! O(0; 0) +
+ " | 1 − x2 − y 2 | 1 x2 + y 2 1
5 ) * m = 0 " / ' x2 +
+ y 2 = 1 6 " # #) * M = 1 "$
6 ! #
fy (x0 ; y0 ) = lim
Δy→0
Δy z
.
Δy
P (x; y)
z = f (x; y) ! " P (x; y)
# x y
$
% % &
fx (x; y), fy (x; y)
zx , zy
∂z ∂z
,
.
∂x ∂y
$ n
n > 2 % % ! '
( u = f (x; y; z)
x P0 (x0 ; y0 ; z0 ) x Δx
" ! &
Δx u = f (x0 + Δx; y0 ; z0 ) − f (x0 ; y0 ; z0 ).
$ u = f (x; y; z) ! x
P0 (x0; y0; z0)
ux (x0 ; y0 ; z0 ) = lim
Δx→0
Δx u
.
Δx
"
) !
% "
* +" "
# !
" ! %
,-.
z =
x2 − y 2
P0 (5; 3).
∂z
2x
x
=
=
;
∂x
2 x2 − y 2
x2 + y 2
∂z
5
5
|P = √
= ;
∂x 0
4
52 − 32
−2y
y
∂z
=
= −
;
2
2
2
∂y
2 x −y
x − y2
∂z
5
|P =
∂x 0
4
∂z
3
|P = − .
∂y 0
4
∂z
3
|P = − .
∂y 0
4
∂z
∂x
z = f (x; y) ! "
z = f (x; y) #
P0 (x0 ; y0 ) Oxy $% $ M0 (x0 ; y0 ; z0 )
& '() $ $ AM0B "
#$ y = y0 *
$ + #
z = f (x; y0) y = y0 , "
df (x; y0 )
= tg α"
"
dx
α - #$ O1X " " +" #$ Ox #"
AM0 B M0 . "
∂z
df (x; y0 )
f (x0 + Δx; y0 ) − f (x0 ; y0 )
=
.
=
lim
dx x=x0 Δx→0
Δx
∂x P0
∂z
$ "
= tg α /"
∂x P0
∂z
P0 (x0 ; y0 )
∂x
Ox
M0 (x0 ; y0 ; z0 )
z = f (x; y) y = y0
0 $
∂z
1
∂x
∂z
∂y
2 # $
+ * " $ #"
# " $
z
Z
y
01
0
y0
M0
αA
B
P0
X
x
∂z
∂x
z = f (x; y)
! "
∂z
∂z
∂
∂ 2z
∂ 2z
∂x
∂x
=
=
= fxy
=
f
(x; y);
2 (x; y);
x
∂x
∂x2
∂y
∂x∂y
∂z
∂z
∂
∂
∂ 2z
∂ 2z
∂y
∂x
=
= fyx
= 2 = fy2 (x; y).
(x; y);
∂x
∂y∂x
∂y
∂y
# u = f (x; y; z) $ %
∂
"
∂
∂u
∂x
∂x
∂
=
∂u
∂x
∂y
∂ 2u
∂ 2u
=
= fxy
= fx2 (x; y; z);
(x; y; z);
2
∂x
∂x∂y
∂u
∂
∂ 2u
∂x
=
= fxz
(x; y; z)
∂z
∂x∂z
&
% % "
n %
(n − 1) '
3
∂ z
∂x∂y
2
z = f (x; y) y
∂ 2z
∂x∂y
∂ 3z
=
∂x∂y 2
∂
∂ 2z
∂x∂y
.
∂y
∂ 2z
∂ 2z
∂ 3z
,
,
∂x∂y ∂y∂x ∂x∂y 2
! "
" z = f (x; y)
#$ %
z = x2y3
& "
∂z
∂z
= 2xy 3 ,
= 3x2 y 2 .
∂x
∂y
' "
∂z
∂ 2z
∂x
=
= 2xy 3 y = 6xy 2 ,
∂x∂y
∂y
∂z
∂
∂ 2z
∂y
=
= 3x2 y 2 x = 6xy 2 .
∂y∂x
∂x
∂
(
∂ 2z
∂y∂x !) *
!
*
∂ 2z
∂x∂y
s
t
∂x
t
= 2cos s ln 2 − sin
∂t
s
t
∂x
= 2cos s
∂s
∂x
∂t
t
1
t 1
= −2cos s ln 2 sin · .
s
s s
∂x
t
∂s
t
t
t
t t
− 2 = 2cos s ln 2 sin · 2 .
ln 2 − sin
s
s
s s
·
!" "
#$%& z = 2x2 − 3y2 − 2xy + 3x − 5y + 1.
2y − 3x
.
#$%' z = 2x
− 5y
#$%$ z = √yx− x .
#$%( z = √x − y.
#$%)* z = y −3 x .
3
x
z = 2y .
#$%))
#$%)+ z = sin
x
√
2 y
√
y
√
.
3
3 x
#$%)# z = tg
#$%), z = arcsin(3y − 2x).
#$%)- z = arctg 3x − 2√y .
#$%)& u = (y)xz .
#$%)' u = xyz .
y = ln(cos u − sin v).
u = (yz)x .
!
z = x2 + y 2 .
z = log3 (x − y 2 ).
" # $! % " $$ % " &
' # ( $$ % ) $
$ % '
$! %
" '
# $! %
*
! ' $
+
, -
#
! $! %
. !
"! $! % !
z = f (x; y) " '
. ! x y ! ( # Δx
Δy - $! % z = f (x; y) ! # Δz
!(# $ ! /
Δz = f (x + Δx; y + Δy) − f (x; y).
+ ,
0 # $! %
Δz #
( $ $! % z = f (x; y)
P (x; y) ! P1 (x + Δx; y + Δy) + 1,
2
*
z = xy2
/ 3 ! $ ! ! + , !
Δz = (x + Δx)(y + Δy)2 − xy 2 = xy 2 + y 2 Δx + 2xyΔy+
+2yΔxΔy + x(Δy)2 + Δx(Δy)2 − xy 2 = (y 2 Δx + 2xyΔy)+
+(2yΔxΔy + x(Δy)2 + Δx(Δy)2).
z
M(x;y;z)
Δz
N
0
M0(x0;y0;z 0 )
Δx
x
P0 (x0;y0)
y
P(x;y)
Δx
Δy
Δz
y Δx+
Δx Δy
+ 2xyΔy
2yΔxΔy + x(Δy) + Δx(Δy)
Δx Δy ! " " #
Δx → 0 Δy → 0 ! "
#
2
2
2
$ % "
" dy
" "
& ' dx " " (
# dx = Δx dy "' '
dy = f (x)dx ) ' "
x z = f (x; y)
' y d z = f (x; y)dx
y * " x + d z = f (x; y)dy
,- .
y = f (x)
x
x
y
z = f (x; y)
y
x y
dz = fx (x; y)dx + fy (x; y)dy.
Δz = f (x+Δx; y +Δy)−f (x; y)!
" Δx = dx Δy = dy # " "!
Δz ""
dz
$ " ω(Δx; Δy) $ ! "
ρ = Δx2 + Δy 2 "% " P (x; y) P1 (x + Δx; y + Δy)&
Δz = dz + ω(Δx; Δy),
lim
ρ→0
ω(Δx; Δy)
= 0.
ρ
P (x; y)
' (
P (x; y)
(
)!
"! " *
fx (x; y)
fy (x; y)
z = f (x; y)
+% (! $ % ! $ !
! (
(
$
(
,
! %!
!
" .
"!
, (
-
)
!
" "! " "
"
"
∂z ∂z
∂x ∂y
z =
P (x; y)
P (x; y)
!"
= f (x; y)
/ (
"
, 2 !
0, "
,
"!
u = f (x; y; z)
0, $ 1
"
Δu
%
"
Δu =
∂u
∂u
∂u
Δx +
Δy +
Δz + ω(Δx; Δy; Δz)
∂x
∂y
∂z
3
ω
lim = 0 ρ = Δx2 + Δy 2 + Δz 2
ρ→0 ρ
du =
∂u
∂u
∂u
dx +
dy +
dz.
∂x
∂y
∂z
z
= xy 2
∂z
∂z
dx+ dy !
∂x
∂y
∂z ∂z
" # $ #
% #
∂x ∂y
∂z
∂z
= (xy 2 )x = y 2 ,
= (xy 2 )y = 2xy.
∂x
∂y
dz =
& " " $ ''('
)' ) Oxy *
* ) ( ! " dz = y 2 dx + 2xydy +
* , " -( ! ' ) ./
- )' z = f (x; y) ") P0 (x0 ; y0 )
dz = fx (x0; y0 )Δx + fy (x0 ; y0)Δy dz = fx (x0; y0 )(x − x0 ) +
+fy (x0 ; y0 )(y − y0 )
0 (! ) 1 ) $ " ) -
2
) M0 M1 KM2
z − z0 = fx (x0 ; y0 )(x − x0 ) + fy (x0 ; y0 )(y − y0 ),
3 z 4
) ") K ) - ) )-)
" *# ( 1 - # "
5 ) 1 $ ) # #
! (
) ) - ) KN
2 0 *
$ )(" ' 3 " )
6 "
!
) ) - ) KN - 3 ' " ! ' ) MN 2
z
M(x;y;z)
K
dz
N
M2
0
M0(x0;y0;z 0 )
M1
Δx
x
P0 (x0;y0)
P(x;y)
y
Δx
Δy
Δz
z = f (x; y)
Δz = fx (x; y)Δx + fy (x; y)Δy + ω(Δx; Δy).
! ω(Δx; Δy) " # ρ = (Δx)2 + (Δy)2
$ ρ# Δx Δy# % ω(Δx; Δy)
&
'() *+
Δz ≈ fx (x; y)Δx + fy (x; y)Δy,
, z = f (x; y)#
Δz = f (x + Δx; y + Δy) − f (x; y).
$ Δz '() *+#
f (x + Δx; y + Δy) − f (x; y) ≈ fx (x; y)Δx + fy (x; y)Δy,
f (x + Δx; y + Δy) ≈ f (x; y) + fx (x; y)Δx + fy (x; y)Δy.
! ! "# !!! ! P (x+Δx; y +Δy)$
! P (x; y)$ ! ! ! "# !
! P (x; y)
% & ! " ! "# n !!!
n > 2 '!$ n = 3
f (x + Δx; y + Δy; Δz + Δz) ≈
(
≈ f (x; y; z) + fx (x; y; z)Δx + fy (x; y; z)Δy + fz (x; y; z)Δz.
√
1
.
2.952 + 4.012
1
) ! * ! !+ ) "#, z =
-!
x2 + y 2
"
. "# +
1
1
≈
+ zx Δx + zy Δy.
2
2
2
(x + Δx) + (y + Δy)
x + y2
'! ! !+
x
y
; zy = − 2
.
zx = − 2
2
3/2
(x + y )
(x + y 2 )3/2
- !! x = 3$ Δx = −0,05$ y = 4$ Δy = 0,01 - +
1
4
3
1
≈ + √ 0, 05 − √ 0,01 ≈ 0,21.
2
2
5 5 5
5 5
2, 95 + 4, 01
/!$ ! . !! *! ,
1
0, $
≈ 0,201 1
2,952 + 4,012
! ,! !!!! ""!!# &
0! "#
2 z = f (x; y) 3 "# ! !!! x y $ !!
!! ! ! ! & $
""!!# & ""!!#$ ! ""!!
# & + d(dz) = d2 z
dx dy
x y
∂ ∂z
∂z
dx +
dy dx+
∂x ∂x
∂y
∂z
∂ ∂z
dx +
dy dy =
+
∂y ∂x
∂y
∂ 2z
∂ 2z 2
∂ 2z
∂ 2z 2
dxdy
+
dxdy
+
dx
+
dy .
=
∂x2
∂x∂y
∂y∂x
∂y 2
d2 z = d(dz) = d
∂z
∂z
dx +
dy
∂x
∂y
=
! " # #
∂ 2z 2
∂ 2z 2
∂ 2z
dxdy
+
d2 z =
dx
+
2
dy .
∂x2
∂x∂y
∂y 2
$ %# & ' % (
'
)
d3 z =
∂ 3z 3
∂ 3z
∂ 3z
∂ 3z 3
2
2
dx
dx
+
3
dy
+
3
dxdy
+
dy .
∂x3
∂x2∂y
∂x∂y 2
∂y 3
* % % # "+ ' n(%
'
&
" ,-
n
∂ nz
Cni i n−i dxi dy n−i .
dn z =
./
∂x ∂y
i=0
' z = f (x; y) #0 %(
) ' - '
(
t : x = x(t), y = y(t) 1% z & '
t # - )
∂z
∂z
& ' dz
#
(
dt
∂x
∂y
dx
dy
2 ) # " % #
dt
dt
' x = x(t) y = y(t) - # t '
z = f (x; y) -+ # (x; y) (
'
t # + Δt3 %
x y # - + Δx Δy
' z 4 + Δz 1 ' z & -
Δz
∂z
∂z
Δx +
Δy + ω(Δx; Δy),
∂x
∂y
ω
! lim = 0 " ρ = Δx2 + Δy 2 # $ !
ρ→0 ρ
Δt % & ' Δt → 0 !
Δz =
lim
Δt→0
Δz
Δx ∂z
Δy
ω
∂z
=
lim
+
lim
+ lim
.
Δt→0
Δt→0
Δt→0
Δt
∂x
Δt
∂y
Δt
Δt
(
) ' * $ &% * ! + "
&* * ! + "
dz
Δx
dx
Δy
dy
=
=
$ & , lim
lim
Δt→0 Δt
dt
dt Δt→0 Δt
dt
, *
ω
ω ρ
= lim
·
lim
Δt→0 Δt
Δt→0
ρ Δt
#
= lim
Δt→0
ω
ρ
· lim
.
ρ Δt→0 Δt
!
Δx2 + Δy 2
ρ
= lim
=
Δt→0 Δt
Δt→0
Δt
2
2
2
Δx
dx
Δy
= lim
+
=
+
Δt→0
Δt
Δt
dt
lim
-
dy
dt
2
.
dx
dt
Δt → 0
' ' ' $
dy
ω
. !
' $ ! lim = 0
!
Δt→0 ρ
dt
ω
ω
' ρ → 0 ,
" lim = lim = 0
ρ→0 ρ
Δt→0 ρ
2
2
ω
dx
dy
=0·
lim
+
= 0.
Δt→0 Δt
dt
dt
/! & +
(
$
dz
∂z dx ∂z dy
=
+
.
dt
∂x dt
∂y dt
3
y= t
dz
dt
z = y x x = cos t
dz
dy
dx
= (y x ) ·
+ (y x )y
= −y x ln y sin t + 3t2 xy x−1 =
dt
dt
dt
= −t3 cos t ln t3 sin t + 3t2 cos t · t3(cos t−1) = t3 cos t−1 (3 cos t − 3 ln t sin t) =
= t(sin t)t
2 −1
(t cos t + 2 sin t · ln sin t).
z = f (x; y) ! y = y(x)
"# z #$ $ x : z = f (x; y(x))
% $ !# #&#' ( $ t )*
x +
dz
∂z dx ∂z dy
=
+
.
dx
∂x dx ∂y dx
,
dx
= 1
dx
-
dz
∂z ∂z dy
=
+
.
dx
∂x ∂y dx
. !$
!$ / - $ & !#& z
∂z
x 0# /
1 !# #!/ *
∂x
&/ z = f (x; y) /# 2& y !
dz
x . ( !#
' ! !$ *
dx
& !# 3$ #$ $
z = f (x; y(x)) % !# & 2# &! $ *
!#$
+# 3 z = f (x; y) ( x = x(u; v)
y = y(u; v) 4)# z 3 #!/ !
&/ *
∂z
∂z
&/ u v , $# & !#&
- $ 3$
∂u
∂v
F (x; y)
x
M (x ∈ M)
y x
M
! y = ϕ(x)
y = ϕ(x)
! "! !#"!
F (x; ϕ(x)) = 0,
"
" x "!
M $!
% ! & !
y = f (x) '
( !" !) y !"
% " ""! * + #
( !) !" !) y ,
y = log3 (x3 + 1).
-.
/!
0!" $! ! # "
!'
+ 1 !#"!
! ! 2 + % " "!
"!'
( + "! y # -. &
3log3 (x
3 +1)
− x3 − 1 = x3 + 1 − x3 − 1 = 0.
% ! "&
# & 2 x ∈ M "! !"! !
") & y ! 23
" "! " x '
2 $! ! ")
x2 + y 2 − 1 = 0 '
!
! # "!)
(
x2 + y 2 − 1 = 0 !" !) y ,
y=
√
1 − x2 ,
√
y = − 1 − x2 .
4 "! !) &! " 2
"! !)
$!
2 2 #
4 '
3y − 3y + x3 − 1 = 0
! 2 2 y ! "3"! 2! & x
y ! 23 2 x = 0 y = 05
x=1 y=1
F (x; y) = 0
x2 + y 2 + 1 = 0
x y
!
y
x
"#
! #
" "
$
y%
F (x; y) = 0
&
F (x; y)
!
!
'
#
#
'
"
()* F (x; y)
Fx (x; y) Fy (x; y)
P0 (x0; y0) F (x0; y0) = 0 Fy (x0; y0) = 0
F (x; y) P0 (x0; y0)
y = y(x)
x0
y(x0 ) = y0
+
, " !
,
!
/
-().)
" #
!
y = y(x) " P0 (x0 ; y0 ) #
F (x; y(x)) ≡ 0 x
/ !
! !
dF
=0
dx
"
-().0
dF
∂F
∂F dy
=
+
dx
∂x
∂y dx
!
∂F
∂F dy
+
=0
∂x
∂y dx
∂F
dy
= − ∂x .
∂F
dx
∂y
,
-()*.
" 1 ! " "
! "
y x3 − 3x + y2 − xy − 1 = 0
P (1; 1)
F (x; y) = x2 −2x+3y 2 −xy−1 = 0
∂F
∂F
= 3x2 −3−y
= 2y−x !"
∂x
∂y
∂F
dy
3x2 − 3 − y
3x2 − 3 − y
= − ∂x = −
=
.
∂F
dx
2y − x
x − 2y
∂y
# $ % & ' x
y =
(
(6x − y )(x − 2y) − (1 − 2y )(3x2 − 3 − y)
.
(x − 2y)2
# (
) & ' * y
y =
(++ ,
(3x2 − 12xy + 3 + y)(x − 2y) − (x − 6x2 + 6)(3x2 − 3 − y)
.
(x − 2y)3
((
+ P (1; 1)
dy
= −6.
x=1 = 1, y | x=1
y=1
dx y=1
- & '* y *
x & (
)
+ +(* + + ( (. Ox 45o tg ϕ = 1"
/+ ( * $ +$ ) ) y =
= f (x) ( ( & 0 & ,
' * $ dy = f (x)dx ((* & (
* *(* x ( ) ) +$ ) +,
) ) x = ϕ(t)
1* +$
(++ 2 &2 u = f (x; y; z; . . . ; t) ( ,
' &)
$ +$
n &2 u = f (x; y; z; . . . ; t) (2 * ( .
∂u
∂u
∂u
∂u
dx +
dy +
dz + · · · +
dt
du =
∂x
∂y
∂z
∂t
( * *.(* x y z . . . t ( & ,
& +$ * 2 &2
z = f (x; y) x y
! "
dz =
∂z
∂z
dx +
dy.
∂x
∂y
dz =
∂z
∂z
du +
dv.
∂u
∂v
# x y
" x = x(u; v) y = y(u; v) $%
z u v &
' ()* +,- ()* +.∂z
∂z ∂x ∂z ∂y
=
+
,
∂u
∂x ∂u ∂y ∂u
/
∂z
∂z ∂x ∂z ∂y
=
+
.
∂v
∂x ∂v ∂y ∂v
∂z
∂z ∂x ∂z ∂y
∂z ∂x ∂z ∂y
+
du +
=
+
dv =
dz =
∂x ∂u ∂y ∂u
∂v
∂x ∂v ∂y ∂v
∂z ∂x
∂z ∂y
∂z ∂y
∂z ∂x
du +
dv +
du +
dv =
=
∂x ∂u
∂x ∂v
∂y ∂u
∂y ∂v
∂z ∂x
∂x
∂z ∂y
∂y
∂z
∂z
=
du +
dv +
du +
dv =
dx +
dy,
∂x ∂u
∂v
∂y ∂u
∂v
∂x
∂y
∂y
∂y
du +
dv = dy.
0 ∂u
∂v
/ dz %
∂x
∂x
du +
dv = dx,
∂u
∂v
dz =
∂z
∂z
dx +
dy
∂x
∂y
x y
1 2 %
3
z = xy 2
dz M(2,00; 1,00) Δx = 0,20
Δz
Δy = 0,10
Δz = f (x + Δx; y + Δy) − f (x; y) = (x +
+Δx) · (y + Δy)2 − xy 2 x = 2 y = 1
M Δz = 2,20 · 1,102 − 2,00 · 1,002 ≈ 0,66
∂z
∂z
dz = dx + dy = y2Δx + 2xyΔy.
∂x
∂y
dx = Δx dy = Δy x y
M
dz = 1,002 · 0,20 + 2 · 2,00 · 1,00 · 0,10 = 0,60.
! Δz dz 0,06
ρ = Δx2 + Δy2 ≈ 0,22.
" Δz ≈ 0,66; dz = 0,60
#$ % dz z = x3y2
∂z
∂z
dx +
dy = 3x2 y 2 dx + 2x3 ydy.
∂x
∂y
dz = 3x2 y 2 dx + 2x3 ydy.
dz =
"
#$ #
1,012 · 0,983 .
&' ( )#$ *+
,, dz , z = x2y3 x = 1
y = 1 Δx = 1,01 − 1 = 0,01 Δy = 0,98 − 1 = −0,02 - .
1,012 · 0,983 f (x + Δx; y + Δy)
dz =
∂z
∂z
dx +
dy = 2xy 3 Δx + 3x2 y 2 Δy =
∂x
∂y
= 2 · 1 · 13 · 0,01 + 3 · 12 · 12 (−0,02) = −0,04.
- (!/ Δz ≈ dz
f (x + Δx; y + Δy) − f (x; y) ≈ dz ⇒ f (x + Δx; y + Δy) ≈ f (x; y) + dz
0 f (x; y) = 12 · 13 = 1
1,012 · 0,983 ≈ 1 − 0,04 = 0,96.
∂z dz
∂x dx
x
z = 2y
y = sin x
∂z
∂x
dz
! "
dx
x
∂z
1
= 2 y ln 2 ·
∂x
y
x
x
dz
1
x
= 2 y ln 2 + 2 y ln 2 − 2 cos x =
dx
y
y
x
x
ln 2
x
sin
2 x
1
x
= 2 sin x ln 2
cos x =
+ 2 sin x ln 2 − 2
(1 − x ctg x) .
sin x
sin x
sin x
dy
dx
#
y = xy
$ y = xy F (x, y) =
dy
yxy−1
= xy − y = 0 % & '()
=− y
dx
x ln x − 1
* " + ! , y
- " x ! y
.
/
" dz
Δx Δy
.
"
z = x2 y 2 ;
/ z = x2 y;
(01 (
(0 z =
Δz
" M(x; y)
M(1; 2);
M(1; 1);
Δx = 0,05;
Δx = 0,01;
Δy = 0,10.
Δy = 0,02.
"
x2 + y 2
.
x2 − y 2
dz
x
z = ln tg .
y
u = xyz.
z = xy x .
z = xey + yex .
z = ey sin x.
z = exy .
! " # $ %& ' "
$ ' #(' ( ) #
# " # $ ( ' " * + #
' (
sin 31◦ cos 61◦ .
! 0,993 · 1,022 .
√
,
3,98 + 2,95.
# -*
.
z = cos t
√
dz
dx
z = s − t t = tg x s = √x
3
√
du
u = xyz x = t + 2 y = et
dt
∂z
dz
∂x
dx
z = yx y = arctg x
v = tg x
* " ' (
dz
z = xy y = arcsin t x = ln t
dt
dz
dx
z = sin(u · v)
dy
dx
y = x + log3 y.
u =
√
x
u P (x; y; z)
l
u
∂u
< 0 u
∂l
∂u
!
∂l
u
"
#
$ % & Δx Δy Δz
P # P1 P = Δl &
& ' ( )*+,
Δx = Δl cos α; Δy = Δl cos β; Δz = Δl cos γ.
(-. )+
/ u
0* (0* -+ ! & Δu P (x; y; z)
Δu = Fx (x; y; z)Δx + Fy (x; y; z)Δy + Fz (x; y; z)Δz + ω,
(-. 0+
! ω ρ =
ω
lim = 0
ρ→0
Δx2 + Δy 2 + Δz 2
ρ
z
γ
p
α
Δy
β
p1
Δz
Δx
l
0
y
x
Δl
Δx Δy Δz
1 &
Δu = Δl u ρ = Δl Δx Δy Δz
Δl u =
Fx (x; y; z)Δl cos α
+ Fy (x; y; z)Δl cos β + Fz (x; y; z)Δl cos γ + ω.
! Δl "# $
Δl → 0%
∂u
Δl u
= lim
= lim (Fx (x; y; z) cos α+
Δl→0 Δl
Δl→0
∂l
ω
.
+Fy (x; y; z) cos β + Fz (x; y; z) cos γ + lim
Δl→0 Δl
& Fx (x; y; z)% Fy (x; y; z)% Fz (x; y; z) # $'
ω
ω
= lim = 0%
# Δl% $ $$ Δl→0
lim
Δl ρ→0 ρ
∂u
= Fx (x; y; z) cos α + Fy (x; y; z) cos β + Fz (x; y; z) cos γ.
∂l
( '
% $
% % # u l
! $) $% %
=
% cos α = 1% cos β = 0% cos γ = 0 % *% ∂u
∂l
= Fx (x; y; z)
+
z = f (x; y)
!"
z = f (x; y) l
∂z
= fx (x; y) cos α + fy (x; y) cos β.
∂l
P0 (5; 3)
+ # z =
l = 3i + 4j
-
3
3
3
=√
= ;
|l|
5
32 + 42
' $ P0 '
cos α =
cos β =
,
x2 − y 2
4
4
= ,
|l|
5
'
' . / 0
,
∂z
3 3
3
53 34
−
= − = .
=
∂l P0 4 5 4 5
4 5
20
P0
|P > 0
z = x2 − y2 ∂z
∂l
0
|P
∂z
∂l
0
=
3
.
20
!"# P (x; y; z)
u = F (x; y; z)
Fx (x; y; z)i + Fy (x; y; z)j + Fz (x; y; z)k.
$ u = F (x; y; z)
grad F (x; y; z) grad F (P ) grad u %
&
grad F (x; y; z) = Fx (x; y; z) + Fy (x; y; z) + Fz (x; y; z),
'!"()
∂u
∂u
∂u
grad u =
+ + .
'!"*)
∂x
∂y
∂z
+ , P (x; y; z)
u = F (x; y; z)
- grad F (P )
!". u = x2 + y2 − z2
P0 (1; 1; 1)
/ 0 1 2 2 P0
∂u
= 2x;
∂x
∂u
= 2;
∂x P0
∂u
= −2z;
∂z
∂u
∂u
= 2y;
= 2;
∂y
∂y P0
∂u
= −2.
∂z P0
3 '!"*)
grad u = 2i + 2j − 2k.
grad u = 2i + 2j − 2k.
l = i cos α + j cos β + k cos γ
l
grad u
u
.
l grad u = ∂u
∂l
u = F (x; y; z)
! " #
$
# % ! $ & ' (
)
grad u = Fx (x; y; z)i + Fy (x; y; z)j + Fz (x; y; z)k.
'
%
!l grad u = grad u · l =
= Fx (x; y; z) cos α + Fy (x; y; z) cos β + Fz (x; y; z) cos γ =
$
∂u
,
∂l
∂u
+
∂l
% # # ! # u = F (x; y; z) ' % !
l %
% +
! " # grad u
# ! # u = F (x; y; z) !
l
, % ϕ %+$ $ %
% l grad u
- $ !l grad u = | grad u| cos ϕ ' % . %
* #
! $ # ! ! &
∂u
= | grad u| cos ϕ.
/
∂l
0 !
l grad u ! $ & (ϕ = 0)
!
∂u
% $ 1
$ # ! ! &
∂l
| grad u|
- % % % ! ( $ % $&2% $ grad u
, &$ $
grad u . " # ! # u = F (x; y; z)
!$# # % % ! %
%
$
% # . " # ! #
3 # % % ! + grad u = grad F (x; y; z) $
P (x0 ; y0 ; z0 ) ! ( # ! ( $#2 '
' ! ( % $
F (x; y; z) = C0
F (x; y; z) − C0 = 0.
z
grad F(P 0 )
r (t 0 )
90
p0
o
L
F(x;y;z)-C 0=0
0
y
x
L
P0 ! "
⎧
⎨ x = x(t),
y = y(t),
⎩ z = z(t),
# x(t)
y(t) z(t) $ %%& ' % & t ( x0 = x(t0 )
y0 = y(t0 ) z0 = z(t0 ) ) * + r = (x; y; z)
, + , %- r = r(t) . +
', ' (x(t0); y(t0); z (t0)) = r (t0)
+, , L r(t0)
/, + ,r̄ (t0 ) = lim
Δt→0
r̄(t0 + Δt) − r̄(t0 )
.
Δt
0 r̄(t0 + Δt) − r̄(t0) , , L
, P0 (r̄(t0)) # ", , r̄(t0 + Δt)1
r̄(t + Δt) − r̄(t0 )
0
2# Δt → 0
Δt+
2 , L ' x(t) y(t) z(t)
' ' +
+
L + ", 3
F (x(t); y(t); z(t)) − C0 = 0.
t
(C0)t = 0
∂F
∂F
∂F
x (t) +
y (t) +
z (t) = 0.
∂x
∂y
∂z
t = t0
Fx (x0 ; y0 ; z0 )x (t0 ) + Fy (x0 ; y0 ; z0 )y (t0 ) + Fz (x0 ; y0 ; z0 )z (t0 ) = 0.
grad u(P0 ) = Fx (x0 ; y0 ; z0 ) + Fy (x0 ; y0 ; z0 ) + Fz (x0 ; y0 ; z0 )
r (t0 ) = x (t0 )i + y (t0 )j + z (t0 )k,
L! "
#$%!&&'
grad u(P0) = 0! " #$%!&&' (
grad u(P0) r(t0 )
L P0!
" )
*+ !
grad u(P0 ) · r (t0 ) = 0.
u = F (x; y; z)
P0
grad F (P0 )
P0
) ) z = f (x; y) (
grad f (x; y) = fx (x; y) + fy (x; y).
#$%!&,'
∂z
- * ∂l
= | grad z| cos ϕ,
l grad z = ∂z
∂z
∂l
∂l
ϕ . l grad z!
/ (
z = f (x; y) grad f (x0; y0) (
P0 (x0; y0)!
F (x; y; z) = 0,
F (x; y; z) ! P0 (x0; y0; z0)
! " # $ $%
$ " & $ ! P0 " #' (
' ! ! P0 " # $ ) "
) grad F (P0 )
* $ ) F (x; y; z) = 0
! P0
grad F(P 0)
z
P0 (x0;y0 ;z0 )
y
x
+ #
" ' ) !
,)
- ! P0 .
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
/ ! N (A; B; C) & (
" $) ) 0 (
) .
Fx (x0 ; y0 ; z0 )(x − x0 ) + Fy (x0 ; y0 ; z0 )(y − y0 ) + Fz (x0 ; y0 ; z0 )(z − z0 ) = 0.
P0
P0 !"#$
x − x0
y − y0
z − z0
=
=
.
m
n
p
% & s(m; n; p) & '
( $
y − y0
z − z0
x − x0
=
=
.
")#
Fx (x0 ; y0 ; z0 )
Fy (x0 ; y0 ; z0 )
Fz (x0 ; y0 ; z0 )
!
z = x2 + 2y2 P0 (1; 1; 3)
* + $ ,+
+ 2y 2 − z = 0 , F (x; y; z) = x2 + 2y 2 − z
- grad F (P0 )$
Fx (x; y; z) = 2x;
Fy (x; y; z) = 4y;
"!#$ x2 +
Fz (x; y; z) = −1.
Fx (P0 ) = 2; Fy (P0 ) = 4; Fz (P0 ) = −1.
%
$
" #
2(x − 1) + 4(y − 1) − 1(z − 3) = 0,
$ 2x + 4y − z − 3 = 0.
% ")# $
x−1
y−1
z−1
=
=
.
2
4
−1
y−1
z−1
x−1
=
=
.
. $ 2x + 4y − z − 3 = 0
2
4
−1
* '
& z = f (x; y) !#
/ ( 0 (
,+
−f (x; y) + z = 0.
, F (x; y; z) = −f (x; y) + z
"1#
gradF (P0)
Fx (x0 ; y0 ; z0 ) = −fx (x0 ; y0 );
Fy (x0 ; y0 ; z0 ) = −fy (x0 ; y0 );
Fz (x0 ; y0 ; z0 ) = 1.
−fx (x0 ; y0 )(x − x0 ) − fy (x0 ; y0 )(y − y0 ) + (z − z0 ) = 0,
z − z0 = fx (x0 ; y0 )(x − x0 ) + fy (x0 ; y0 )(y − y0 ).
x − x0
y − y0
z − z0
=
=
.
−fx (x0 ; y0 )
−fy (x0 ; y0 )
1
!
" P0 # $!
% & ' ( ! # )
% #
F (x; y; z) = 0 "
Fy (P0 )
Fz (P0 )
Fx (P0 )
; cos β =
; cos γ =
,
| grad F (P0 )|
| grad F (P0 )|
| grad F (P0 )|
*
2
2
2
| grad F (P0 )| =
(Fx (P0 )) + (Fy (P0 )) + (Fz (P0 )) , #)
% z = f (x; y)- . !- " $ )
cos α =
+
−fy (x0 ; y0 )
1
−fx (x0 ; y0 )
; cos β =
; cos γ =
,
| grad F (P0 )|
| grad F (P0 )|
| grad F (P0 )|
/
| grad F (P0 )| = (fx (x0 ; y0 ))2 + (fy (x0 ; y0 ))2 + 1
cos α =
+
2
z = x2 +2y
o
M(1; 1)
Ox 60
0 1 ! cos α = cos 60o = 21 cos β = cos(90o − 60o ) = cos 30o =
√
3
2
cos β
cos2 α + cos2 β = 1
M
∂z
|M = 2x|x=1 = 2;
∂x
∂z
|M = 4y|y=1 = 4.
∂y
√
√
∂z
3
∂z
∂z
1
|M =
|M cos α +
|M cos β = 2 + 4
= 1 + 2 3 ≈ 4,46.
∂l
∂x
∂y
2
2
XOY M(1; 1)
!
" z = x2 + 2y2 # $ %
XOY
77o
&'( z = 2x2 +y2
M(1; 1)
O(0; 0)
) * +
MO +
, - .
" MO = (0 − 1; 0 − 1) = (−1; −1)
+
MO
.
|MO|
√
|MO| = (−1)2 + (−1)2 = 2
1
1
1
l MN = − √ ; − √
⇒ cos α = − √ ;
2
2
2
l MN =
1
cos β = − √ .
2
/ 0 +
∂z
∂z
|M = 4x|x=1 = 4;
|M = 2y|y=1 = 2 ⇒
∂x
∂y
1
1
6
∂z
√
√
|M = 4 −
⇒
+2 −
= − √ ≈ −4,3.
∂l
2
2
2
M
! " ! XOY
#$%
z = x2 + 2y2
M(1; 1)
& ' ( ) ! #$*
+ + ! grad z (
grad z|M =
∂z
∂z
|M ; |M
∂x
∂y
= (2; 4).
, - #$
! ' "
" .
∂z
∂x
2
+
2
∂z
/
∂y
M "
√ √
22 + 42 = 20 ≈ 4,47 " '
#$*
#$#
x2 y 2 z 2
+
−
=0
16
9
8
P0 (4; 3; 4)
& ' (
+
x2 y 2 z 2
+
−
F (x; y; z) = 0 "( F (x; y; z) =
16
Fx (x; y; z) =
x
;
8
1
Fx (P0 ) = ;
2
9
8
2y
z
; Fz (x; y; z) = − .
9
4
2
Fy (P0 ) = ; Fz (P0 ) = −1.
3
Fy (x; y; z) =
, %0*1 !
(
1
2
(x − 4) + (y − 3) − (z − 4) = 0
2
3
3x + 4y − 6z = 0.
x−4
y−3
z−4
=
=
.
1/2
2/3
−1
y−3
z−4
x−4
=
=
3x + 4y − 6z = 0
1/2
2/3
−1
! "#$ %
&# ' (!) *! Ox + α
, z = 2x2 + xy + 3y2 M(2; −2) α = −45o
- z = x2 − y2 M(−1; 1) α = 120o
z = xy
z =
M(2; 1) cos α =
1
0 < β < π/2
3
x2 − y 2 M(5; 3) cos α = cos β < 0
! "#$
&# M ' ) . &# # &# N
z = xy1 M(1; 1) N (−1; 1)
z = √x − y M(1; 0) N (0; 1)
u = xy + yz + zx M(2; 1; 3) N (5; 5; 15).
/ , + "#$ %
&# ! . "#$ &#
+ 0*1! #* ( "#$
&#
/ z = 2x3 + 3y3 − 2xy M(2; 1).
z = 2 y2 − x2 M(3; 5).
u = 2xyz M(3; 2; 1).
, u = 4x2 + y2 + z2 M(1; 1; 1).
-
z=
x2 y 2
−
2 3
P0 (2; 3; −1)
!
" # $ % & '
( ' )
* # + ( ! $
, -./0
f (x) = f (x0) +
+
(
f (x0 )
f (x0 )
(x − x0 ) +
(x − x0 )2 + · · · +
1!
2!
,1--/
f n−1 (x0 )
(x − x0 )n−1 + O((x − x0 )n ),
(n − 1)!
! # $ $ n ρ → 02 #
3 (# 2 4
(# 4 5 ( 2 ' " $
, ! / ( * ($$ 5 # 0
O(ρn )
ρ = x − x0
1
fx (x0 ; y0 )Δx + fy (x0 ; y0 )Δy +
1!
1
fxx (x0 ; y0 )Δx2 + 2fxy
+
(x0 ; y0 )ΔxΔy + fyy
(x0 ; y0 )Δy 2 + O(ρ3 ),
2!
ρ = Δx2 + Δy 2 Δx = x − x0 Δy = y − y0
,1-/
f (x; y) = f (x0 ; y0 ) +
(
2
2
6 " 2 # ! # $
$ 2 ' " # 0
f (x; y) = f (x0 ; y0 ) +
+
df (x0 ; y0 ) d2 f (x0 ; y0 )
+
+ ···+
1!
2!
dn−1 f (x0 ; y0 )
+ O(ρn ).
(n − 1)!
,1-7/
8 2 # ,7.7/ n = 2
O(ρn )
m
f (x1 ; x2 ; . . . ; xm ) = f (x01 ; x02 ; . . . ; x0m ) + fx 1 (x01 ; x02 ; . . . ; x0m)Δx1 + !"#"$
+fx 2 (x01 ; x02 ; . . . ; x0m )Δx2 + · · · + fx m (x01 ; x02 ; . . . ; x0m )Δxm + O(ρ2 ),
m
Δxi = xi − x0i i = 1, 2, . . . , m ρ =
Δx2i
i=1
% m m
fi (x1; . . . ; xm ) &
⎧
f (x ; x ; . . . ; xm ) = 0,
⎪
⎪
⎨ 1 1 2
f2 (x1 ; x2 ; . . . ; xm ) = 0,
.........
⎪
⎪
⎩ f (x ; x ; . . . ; x ) = 0.
m 1
2
m
"##
!"#'$
m P0 (a1 ; a2 ; . . . ; am )
!
(
X ! m
$ F
⎛
x1
⎛
⎞
X = ⎝ ⎠ ,
xm
F (X) = ⎝
f1 (x1 ; . . . ; xm )
⎞
⎛
⎠,
0 = ⎝ ⎠
0
fm (x1 ; . . . ; xm)
!"#'$
*
0
⎞
!"#)$
!"#+$
*, - !"#'$ . /
-
% *, - Xn !"#+$
ε -
F (X) = 0.
⎛
xn1
⎞
Xn = ⎝ ⎠ ,
xnm
⎛
ε=⎝
Δx1
Δxm
⎞
⎠,
!"#0$
Δxi = xi − xni xi = xni + Δxi
ε = X − Xn X = Xn + ε.
!
"#
⎧
⎨ f1 (xn1 + Δx1 , . . . , xnm + Δxm ) = 0,
...
⎩
fm (xn1 + Δx1 , . . . , xnm + Δxm ) = 0.
$ % & ' %()
*
' +)
, ) # %
' -
)
fi (xn1 + Δx1 , . . . , xnm + Δxm ) = fi (xn1 , . . . , xnm )+
+
m
fi,x
(xn1 , . . . , xnm )Δxj + α2 ρ2 ,
j
j=1
fi,x
(xn1 , . . . , xnm ) =
j
∂fi (xn1 , . . . , xnm )
∂xj
!
# . )
) / %0 Δxnj ,
(j = 1, 2, . . . , m) % /n & 0 xn1 , . . . , xnm # %
, Δxj
⎧
(xn1 , ..., xnm)Δxn1 + ... + f1,x
(xn1 , ..., xnm)Δxnm = 0,
⎨ f1 (xn1 , ..., xnm) + f1,x
m
1
...
⎩
(xn1 , ..., xnm)Δxn1 + ... + fm,x
(xn1 , ..., xnm)Δxnm = 0.
fm (xn1 , ..., xnm) + fm,x
m
1
.
1 * )
% Δxnj (j = 1, . . . , m)#
/(n + 1) & 0 ' 2
xn+1
= xn1 + Δxn1 , . . . , xn+1
= xnm = +Δxnm .
1
m
xnj
2
3 # )
(j = 1, . . . , m)
"# & # %
% Δxnj
'
+)
) xn+1
%
j
.#
) # (n + 2) # & ! (
& # % % % 4 %
) ,4
)
δ
|Δxnj | < δ j = 1, . . . , m
! "
# $%&'( )
* * * +,
, $%&-(
$%&'( !
F (Xn ) + W (Xn ) · εn = 0,
$%&&&(
" W (Xn ) . ) /
0 ) fi (x1 , . . . , xm )
(xn1 , . . . , xnm)
⎛
⎞
∂f1
∂f1
...
⎜ ∂x1
∂xm ⎟
⎜
⎟
W (X) = ⎜ . . . . . . . . . ⎟ .
$%&&1(
⎝ ∂fm
∂fm ⎠
...
∂x1
∂xm
# $%&&&( ) )
εn = −W −1 (Xn )F (Xn ),
$%&&2(
−1
" W (Xn ) . ) ) /
Xn
3 (n + 1) 0
Xn+1 = Xn + εn .
$%&&%(
%&&
x1, x2, x3 x, y, z
%&& !"
x2 + 4y 2 − 1 = 0
" δ = 0,01
y − x3 = 0
#
,
x1 = x x2 = y f1 (x1 ; x2 ) = x2 + 4y 2 − 1 f2 (x1 ; x2 ) =
= y − x3
∂f1
∂f1
∂f1
∂f1
= 2x,
= 8y,
=
=
x2 + 4y 2 − 1
∂x
∂x
∂x
∂y
1
2
;
F (X) =
∂f2
∂f2
∂f2
∂f2
y − x3
= −3x2 ,
= 1.
=
=
∂x1
∂x
∂x2
∂y
W (X) =
2x 8y
−3x2 1
.
!" #
F (X) W (X)
$ x0 y0 δ%
& #$ $
'(%(% )
2
y2
x2 + 4y2 − 1 = 0 * x1 + (0,5)
= 1
2
a = 1 b = 0,5% ) y − x3 = 0
y = x3% # #
$ + ,% -./
y
0,5
1
-1
x
-0.5
0 #$ 1 $
$ x0 = 0,502
y0 = 0,25%
& $
1 '(%(% ! #
1 %% - %
$
( 3 ,4 .5/6
⎞
1
4y
−
2
2
⎜ 2x + 24x y
x + 12x y ⎟
W −1 (X) = ⎝
⎠.
3x2
x
2x + 24x2 y x + 12x2 y
⎛
⎧
ε = −W −1 (X)F (X)
x2 + 4y 2 − 1 4y(y − x3 )
−
,
2x + 24x2 y
x + 12x2 y
2 2
2
3x (x + 4y − 1)
x(y − x3 )
⎪
⎪
+
.
⎩ Δy = −
2x + 24x2 y
x + 12x2 y
x0 = 0,50 y0 = 0,25
! " Δx0 = 0,30# Δy0 = 0,10
⎪
⎪
⎨ Δx = −
$% !
x1 = x0 + Δx0 = 0,8 y1 = y0 + Δy0 = 0,35
! ! &
% %
! " x1 y1 ! Δx1 =
= −0,084# Δy1 = 0,0014# x2 = x1 + Δx1 = 0,716# y2 = y1 + Δy1 = 0,3514
! ! &
|Δx1 | < 0,01
% %
|Δy1 | < 0,01
'"! % % ( % &!
|Δx0 | < 0,01
|Δy0 | < 0,01
i
xi
yi
Δxi
Δyi
) ) ) ) * ) ) ) )
) +) ) ) )+ ) ))
* ) , - ) ) ))++ ) )).
) ,), ) )
/ & % i = 2 |Δx2| < 0,01#
|Δy2 | < 0,01
0 x3 = 0,71# y3 = 0,35
0,01
* z = f (x; y)
M (x ; y ) ∈ D(f ) x = x
y = y
f (x ; y ) > f (x; y)
(x; y) (x ; y ) δ
! "
0
0
0
0
0
0
0
0
0
z = f (x; y)
M0 (x0 ; y0 ) ∈ D(f )
f (x0 ; y0 ) < f (x; y)
(x; y) (x0; y0)
! "
# "
$ % " &
" ' #
( & x = x0 + Δx y = y0 + Δy )
f (x; y) − f (x0 ; y0 ) = f (x0 + Δx; y0 + Δy) − f (x0 ; y0 ) = Δf (x0; y0 ).
* + Δf (x0 ; y0 ) < 0 " , , "' ,
, "
, f (x; y)
M(x0 ; y0 )
-* + Δf (x0 ; y0 ) > 0 " , , "' ,
, "
, f (x; y)
M(x0 ; y0 )
. " # #
"
,
!" #
z = f (x; y) ! $ x = x0 y = y0 %
$ $ $ z
&
$ !
&
$
" y " /
y = y0 0 f (x; y0 ) # "
x 0 " x = x0 1 *
∂z
∂x
x=x0
y=y0
∂z
∂y
x=x0
y=y0
z
M0
y
P0
x
M0
z
y
P0
x
!
" #
$ %&
& ' ( "
&
$
& & %& %
%
) " # "
∂z
= +2x
* " ' ( z = x2 − y 2 " %
∂x
∂z
= −2y % # "
x = 0 y = 0 + $
∂y
' ( "
%&
&
,! $ ' (
"
# &
&
" %
( %
-
f (x; y)
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
·
−
∂x2
∂y 2
∂ 2 f (x0 ; y0 )
∂x∂y
fyy
(x0 ; y0 ) < 0
f (x; y)
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
·
−
∂x2
∂y 2
∂ 2 f (x0 ; y0 )
∂x∂y
2
>0
∂ 2 f (x0 ; y0 )
< 0;
∂x2
>0
∂ 2 f (x0 ; y0 )
> 0;
∂x2
2
fyy
(x0 ; y0 ) > 0
f (x; y)
2
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
∂ 2 f (x0 ; y0 )
·
−
< 0;
∂x2
∂y 2
∂x∂y
2
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
∂ 2 f (x0 ; y0 )
·
−
= 0
∂x2
∂y 2
∂x∂y
!
" # $ $ !
% & ' (
(x0 ; y0 ) $ fxx
(x0 ; y0 ) fyy
(x0 ; y0 ) # )!
2
* fxx (x0 ; y0 ) · fyy (x0 ; y0 ) − fxy
(x0 ; y0 ) > 0
2
fxy (x0 ; y0 )
< 0.
# fxx
(x0 ; y0 ) < 0 fyy
(x0 ; y0 ) <
(x ; y )
fxx
00
+
z = x3 + y 3 − 3xy !
' ()
! "
#
⎫
∂z
⎬
= 3x2 − 3y = 0, ⎪
∂x
∂z
⎭
= 3y 2 − 3x = 0. ⎪
∂y
$% !
x1 = 1,
y1 = 1
x2 = 0,
y2 = 0.
2
2
∂ z
∂ 2z
∂ z
=
−3,
=
6x,
= 6y.
∂x2
∂x∂y
∂y 2
M(1; 1)
∂ 2z
∂ 2z
∂ 2z
A=
=
6,
B
=
=
−3,
C
=
= 6,
∂x2 x=1
∂x∂y x=1
∂y 2 x=1
y=1
y=1
y=1
AC − B 2 = 36 − 9 = 27 > 0; A > 0( C > 0).
(1; 1) ! "
zmin = −1.
#
M2 (0; 0)
A = 0,
B = −3,
C = 0;
2
AC − B = −9 < 0.
$ %
!
& '
!
()
"
%
#*%+ z = f (x; y)
M
M
M
!
, ! z = f (x; y)
' G
! - ' %
& - - ' ' .
. ( '
' "
' / !% 0 ' . . !
' G -
( - ! z = f (x; y)% 1 '
G
!
z = f (x; y) G
!
"
G #
! $
z = f (x; y)
G
"
#
$
$
$
$ #
%
&' ( #
z = x2 − y 2 x2 + y 2 4
zy
) * +
= −2y ) $
"
2x = 0,
−2y = 0,
zx = 2x
P0 (0; 0) $
+$
# x2 + y 2 = 4 , $ #
z = x2 − y 2 #
$
$ x : z = x2 − (4 − x2) z = 2x2 − 4 % −2 x 2 -
# $
$
# x2 +y 2 = 4 #
$
$ $ z = 2x2 − 4
[−2; 2] +$
(−2; 2)
- z = 4x 4x = 0
x = 0. z|x=0 = −4 , z|x=−2 = 4 z|x=2 = 4 /
&
−4
z
z=f(x;y)
y
L
x
!!" # $ %
#&' y '
x '
!! # y ' &
x
( $' ' $ # ' # !!"
) * +#& %
,
- y x' !!" '
u . x' ' &
u x
' /0$ $
du
∂f
∂f dy
=
+
⇒
dx
∂x ∂y dx
∂f
∂f dy
+
= 0.
∂x ∂y dx
!!1
2 / !!" x' & 3
∂ϕ ∂ϕ dy
+
= 0.
∂x
∂y dx
( # & x y '
!!"
!!4
0& %
λ
!"#
$
%& ! '
∂f
∂ϕ
+λ
∂x
∂x
+
∂f
∂ϕ
+λ
∂y
∂y
dy
= 0,
dx
()
$"# $ !
* λ & $ ()
+ !+'
,- .
()'
∂f
∂ϕ
+λ
= 0.
∂y
∂y
$ x y !
!"
∂f
∂ϕ
+λ
= 0.
∂x
∂x
, .& ! $" $ ! $'
⎧
∂f
∂ϕ
⎪
⎪
⎪ ∂x + λ ∂x = 0,
⎨
∂ϕ
∂f
+λ
= 0,
⎪
⎪
∂y
∂y
⎪
⎩
ϕ(x; y) = 0.
(
, ( $$ $ ! ! -
! / &
! () $$" $
. ! 0 -
F (x; y; λ) = f (x; y) + λϕ(x; y)
((
x& y λ
, .& $ $ ! - ! !
1 ! 2 0 - !
+ +!" ! " ((& $ !"
. x& y λ& . ! (
+ . x y - + + λ ! /
& +! ! $ ( $$" $ !
& ! $ +
3- 4 . ! $ ! +
. . - .
5 $ $ ! "-
3xyz
λ = −
2a
⎧
3x
⎪
⎪
yz
1
−
(y
+
z)
= 0,
⎪
⎪
2a
⎪
⎪
⎪
⎪
⎨xz 1 − 3y (x + z) = 0,
2a
⎪
⎪
3z
⎪
⎪
(x
+
y)
= 0,
xy
1
−
⎪
⎪
2a
⎪
⎪
⎩
xy + xz + yz − a = 0.
x y z
⎧
3x
⎪
⎪
1 − (y + z) = 0,
⎪
⎪
2a
⎪
⎪
3y
⎨
1 − (x + z) = 0,
2a
⎪
3z
⎪
⎪
(x + y) = 0,
1
−
⎪
⎪
2a
⎪
⎩
xy + xz + yz − a = 0.
x = y !
y = z
x=y=z=
a
.
3
"
"# " x = 0
y = 0 z = 0
$ $"# "
"!% " &
" "
a
3
' (
!
"!%
) %
! "!%
&*
z = f (x; y)
P0 (x0; y0)
grad f (P0 )
! P1 (x1; y1) grad f (P1)
"#$ "# %
grad f (P1) P2 (x2; y2)
& ' (
# " " | grad f (Pn)|
$
)
" & $
'
*
! +
, $ -
$
./ /
− 6x − 3y.
z = x2 + xy + y2 −
0 1 ( $
$
! +
$1
⎧
⎪
⎨
⎪
⎩
∂z
∂z
= 2x + y − 6;
= x + 2y − 3
∂x
∂y
∂z
= 0,
2x + y − 6 = 0,
x = 3,
∂x
⇔
⇔
∂z
x
+
2y
−
3
=
0
y = 0.
=0
∂y
( $
2
· zyy
− (zxy
)
Δ = zxx
zxx
= 2;
zyy
= 2;
zxy
= 1 ⇒ Δ = 2 · 2 − 1 = 3 > 0.
! $
M(3; 0) $ ! 2 zxx
> 0+
!
31 4 M(3; 0) $
− 12x − 15y
z = y3 +3x2y −
zx
= 6xy − 12; zy = 3y 2 + 3x2 − 15
2
zx = 0,
x + y 2 = 5,
6xy − 12 = 0,
⇔
⇔
⇔
2
2
zy = 0
xy = 2
3y + 3x − 15 = 0
2
x + 2xy + y 2 = 9,
(x + y)2 = 9,
x + y = ±3,
⇔
⇔
⇔
2
2
x − y = ±1.
x − 2xy + y = 1
(x − y)2 = 1
M1 (1; 2);
M2 (2; 1);
M3 (−1; −2);
M4 (−2; −1).
2
Δ = zxx
zyy − (zxy
)
! "
zxx
= 6y; zyy
= 6y; zxy
= 6x ⇒
2
Δ = 6y6y − (6x) = 36(y 2 − x2 )
Δ|M1 = 36(4 − 1) > 0; Δ|M2 = 36(1 − 4) < 0
Δ|M3 = 36(4 − 1) > 0; Δ|M4 = 36(1 − 4) < 0.
# $ " M1 M3 $ ! " M2
M4 %
zxx
" M1 M3
zxx
|M1 = 6 · 2 > 0;
zxx
|M3 = 6 · (−2) < 0.
# $ M1 &
'
(
z = 2x2 + (y − 1)2
M3
%
)" % !
∂z
= 4x;
∂x
∂z
= 2(y − 1)
∂y
⎧
∂z
⎪
⎨
= 0,
x = 0,
4x = 0,
∂x
⇔
⇔
∂z
y = 1.
2(y
−
1)
=
0
⎪
⎩
=0
∂y
Δ = zxx zyy − (zxy )2
zxx
= 4,
zyy
= 2,
zxy
= 0 ⇒ Δ = 4 · 2 − 0 = 8 > 0.
M(0; 1)
Δzxx > 0
! " M(0; 1) #
$% $
z = 1 − x2 − y 2
x + y − 1 = 0
& ' ( ) ) z = 1 − x2 − y2
) * +,- L : x + y − 1 = 0
Oxy
z
N
z= 1-x 2-y
M
P
A
2
0
x+y-1=
B
y
x
. / 0
1 1
/ ' ) P 2 ; 2 0
1 0 A(1; 0) B(0; 1) '
2 A B
3 2 M
/ 2 N
. x + y − √1 = 0 y = 1 − x !
z = 1 − x2 − (1 − x)2 ⇔ z = 2x − 2x2 !/ x
2x − x2 0 ⇔ x ∈ [0; 1]
zx =
2 − 4x
√
2 2x − 2x2
z 12 =
1
zx = 0 ⇔ x0 =
2
1
= √ z(0) = 0 z(1) = 0
2
! " !
1
1
1
1
=√
x 0 = ⇒ y0 = 1 − x 0 = ⇒ z
2
2
2
2
x1 = 0 ⇒ y0 = 1 − x0 = 1 ⇒ z(0) = 0
x2 = 1 ⇒ y0 = 1 − x0 = 0 ⇒ z(1) = 0#
$%#& z = 6 − 3x − 4y
x2 + y2 = 1
' ( ) *"
F (x; y; λ) = 6 − 3x − 4y + λ(x2 + y 2 − 1).
!
⎧
3
⎪
⎧
⎧
⎪
x=
,
⎪
⎨
⎨ Fx = 0,
⎨ −3 + 2λx = 0,
2λ
2
F = 0, ⇔
−4 + 2λy = 0,
⇔
⇔
⎪ y= ,
⎩ y
⎩ 2
⎪
x + y 2 − 1 = 0.
Fλ = 0.
λ
⎪
⎩ 2
x + y 2 − 1 = 0.
⎧
⎧
3
3
⎪
⎪
⎪
x
=
,
x=
,
⎪
⎪
⎪
⎪
2λ
⎪
⎪
2λ
⎨
⎨
2
2
y= ,
⇔
⇔
y= ,
⎪
⎪
λ
λ
2
⎪
⎪
⎪
⎪
2
⎪
⎪
5
2
25 − 4λ
⎩
⎪
⎩
λ=± .
− 1 = 0.
= 0.
2
2
λ
4λ
⎧
⎧
3
3
⎪
⎪
x2 = − ,
x1 = ,
⎪
⎪
⎪
⎪
⎪
⎪
5
5
⎨
⎨
4
4
⇔
y1 = ,
y2 = − ,
⎪
⎪
5
5
⎪
⎪
⎪
⎪
⎪
⎪
5
⎩
⎩ λ = 5.
.
=
−
λ
1
2
2
2
⎧
3
⎪
x=
,
⎪
⎪
⎪
2λ
⎪
⎨
2
y= ,
⇔
λ
⎪
2
⎪
⎪
⎪
3
⎪
⎩
+
2λ
d2F = Fxx dx2 + 2Fxy dxdy + Fyy dy2
Fxx
= 2λ; Fyy
= 2λ; Fxy
= 0 ⇒ d2 F = 2λdx2 + 2λdy 2
5
3
4
λ1 = 2 x1 = 5 y1 = 5 d2F = 5(dx2 + dy2) > 0
λ2 = − 52 x2 = − 35 y2 = − 45 d2F = −5(dx2 + dy2 ) < 0
4
−4
= 1,
5
4
3
z = 6 − 3 −
−4 −
= 11.
5
5
z = 6 − 3
3
5
! " #z $ ! " #z $
z = 6 − 3x − 4y % x2 + y2 = 1
&' M − 35 ; − 45
M
3 4
;
5 5
.
()*
z = x2 − xy + y2 − 4y − x x 0 y 0 3x + 2y − 12 0
+ , ' - % . %
zx = 0,
⇔
zy = 0.
/
2x − y − 1 = 0,
⇔
−x + 2y − 4 = 0.
2 > 01 3 > 01 3 · 2 + 2 · 3 − 12 = 0
M(2; 3)
x = 2,
y = 3.
0 '
- . % z|M = 22 − 2 · 3 + 32 − 4 · 3 −
% 0 / x = 0
− 2 = −7 2 . %
zI = y 2 − 4y 2y − 12 0 y 0 -
,
0 ,
. % [0; 6]
- 3 (zI )y = 2y−4
(zI )y = 0 ⇔ 2y−4 = 0 ⇔ y = 2 -
. %
zI |y=2 = 22 − 4 · 2 = −4 -
. %
% '
zI |y=0 = 0 zI |y=6 = 62 −4·6 = 12
0 ,
x = 0 . % y = 6 : (zI ) = 12 , 4
y = 2 : (zI ) = −4
y = 0
y = 0 x = 4 : (zII ) = 12
1
1
x = : (zII ) = −
2
4
3x + 2y − 12 = 0
! " y = 6 − 3x
#
2
19x2
− 19x + 12
zIII =
4
x 0,
x 0,
x 0,
3x
⇔
⇔
y 0.
x 0.
0.
6−
2
x
!
"
zIII [0; 4] #!
"
x = 2 y = 3 : zIII =
= −7 ! $ x = 0 y = 6 : zIII = 12 %& '
$
( !) *
!
&)!
) "
z = 12 ! "
M1 (0; 6)
M2 (4; 0) z = −7 $ M3 (2; 3)
+, z = 12 z = −7
!
& - .$ - /
) "
z = x + xy + y − x − 2y
z = x4 + y4 − 2x2 + 4xy − 2y2
z = y2 − 2x2 − 2y + 1
2
- .
2
'
º
- 0
º
º
- /
& - - $ - -- ) ' )
1) "
- -
- --
z = 2x + y
z = x2 + y 2
x2 + y 2 = 5
2x + 3y = 6
& - -2$ - -3
"
!
!
z = xy + x + y
- -2
1 x 2, 2 y 3.
z = xy
x2 + y 2 1.
z = x2 + y 2 z = 2x + y
M
!" # $
" % & ' ( $!
)
*
+, ( $
-
. w = f (z)
z M
z ∈ M
w ! " w = f (z)
# $
M
% " % N
w " %
/ $ z - 0 w
1 , 0 2 w = f (z) 3 , $ (3 3 M z 3
N w
.. & " w = f (z) M
z w
4 0 $
+ ( z = f −1 (w)0 (!
32+ 3 N w 3 M z
2+ 3 $
z
w
n ∈ N w = z n
w = ρ(cos θ + i sin θ)
!
|z|n
z = r(cos ϕ+i sin ϕ)
ρ=r ,
θ = nϕ.
w = zn
z
( ($
'
n
" #
'
$ %&
(n − 1) arg z
k 2π
< ϕ < (k + 1) 2π
n
n
! w
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k = 0, 1, . . . , n − 1
(
'
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w=
√
n
z ) n&
M '
(% %
!
√
n
z
n
√
w = n z
n
,
x = 0 y = ϕ
( ** +'
$
w = ez
z = x + iy
'
!
z = 0 '&
(
ez = ex (cos y + i sin y)
&
-(
**.*
eiϕ = cos ϕ + i sin ϕ.
-
!
e2πki
2πi /!
= cos 2πk + i sin 2πk (k ∈ Z) !
ez+2πki = ez · e2πki = ez · 1 = ez .
/
w
!
1 '
z ' ! ln z
ew = z -0
% &
w = ln z
! !
z = 0
z % ' &
( !( &
2π
ln z = ln r + i(ϕ + 2πk), k ∈ Z.
ϕ = 0 r = 0
ln z k = 0
!" # # ln z ln z
k ln z $
ln z # z = 0
% & ' #& '
' ' w = za a = α + iβ ( za = ea ln z
)* + ln(−1).
, ( - ! + −1 = 1(cos π + i sin π) z = −1 |z| = r = 1 ϕ = π .
ln z ln(−1) = ln 1 + i(π + 2πk) = i(π + 2πk), k ∈ Z.
! # ln(−1)
sin z cos z
−iz
−iz
$ sin z = e −2ie / cos z = e +2 e .
0 1
' ##& 2
- eiz = cos z + i sin z
3 " ' 44
2π sin(−z) = − sin(z) cos(−z) = cos(z) sin2 z + cos2 z = 1 sin 2z =
= 2 sin z cos z cos 2z = cos2 z − sin2 z 5 sin z
cos z #
)* * cos i
, ( - .
iz
2
cos i =
iz
2
ei + e−i
e−1 + e1
=
≈ 1,54.
2
2
! # ( +
)* 4
w = f (z) z = x + iy w =
= u(x; y)+iv(x; y)
z0 = x0 + iy0 lim f (z)
z→z0
lim f (z) = x→x
lim u(x; y) + i x→x
lim v(x; y).
z→z0
0
y→y0
0
y→y0
w = f (z)
z0
z→z
lim f (z) = f (z0 ) f (z)
D
0
f (z) f (z)
f (z + Δz) − f (z)
.
Δz→0
Δz
$ f (z)
f (z) = lim
! " #
% z
& ' ($
% )
z$ w = f (z)
* ! f (z) = z2 − √z+3ez −
− 5 ln z + sin z + cos z + tg z
+ , -
1
1
5
.
f (z) = 2z − √ + 3ez − + cos z − sin z +
z
cos2 z
2 z
! ( w . u = u(x; y) %
. v = v(x; y)$ w = u + iv
( .
/ "# $%&'() * f (z) =
= u(x; y) + iv(x; y) z
u(x; y) v(x; y) (
( f (z) z +(
⎧
(
∂u
∂v
⎪
⎨
=
,
∂x
∂y
∂u
∂v
⎪
⎩
=− .
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∂x
' /)
∂v
∂v
∂u
∂u
∂u
∂v
∂u
∂u
+i
=
−i
=
−i
=
+i .
∂x
∂x
∂y
∂y
∂x
∂y
∂y
∂x
f (z) =
f (z) = z̄
#$ %% !
!
z̄ = z − iy
z = x + iy
"
∂v
∂u
∂v
∂u
= 1;
= −1 ⇒
=
,
u(x; y) = x, v(x; y) = −y →
∂x
∂y
∂x
∂y
&
' #
⇒
' #
f (z) = z̄
'"
( f (z)
D
!
)
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+
+
+ +
$
+
+
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y,
!
*
+ ! +
+ .
+, !
2
v(x; y)
!
2
"
x,
2
∂ u ∂ u
+
= 0.
∂x2 ∂y 2
12!
, $ ! $
∂ 2u
∂ 2v ∂ 2u
∂ 2v
=
/
=−
2
2
∂x
∂x∂y ∂y
∂x∂y
0 !
*
+ ! $ ! + ' #
%
2
∂ v ∂ v
+
= 0.
∂x2 ∂y 2
3 %, $ 4 , ' "
#,
6
$5
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0
! 7
, $ 2 ! "
, +
! ! +
2 ! ' #
+ ! * *,
4 ,
888 ! 5 2
= x3 − 3xy 2
ϕ(x; y) = x y ψ(x; y) =
3 2
ϕ(x; y)
ψ(x; y)
!
∂ϕ
= 3x2 y 2
"
# $%
# % &
∂x
∂ 2ϕ
∂ϕ
∂ 2ϕ
∂ 2ϕ ∂ 2ϕ
= 2x3 y
= 6xy 2
= 2x3
+ 2 = 6xy 2 + 2x3 = 0
2
2
∂x
∂y
∂y
∂x2
∂y
' ϕ(x; y) = x3 y 2 (
!
)
( "( ψ(x; y) *
∂ψ
∂ 2ψ
∂ψ
∂ 2ψ
∂ 2ψ ∂ 2ψ
= 3x2 − 3y 2 +
= −6xy +
= 6x+
= −6x
+
=
2
2
∂x
∂x
∂y
∂y
∂x2 ∂y 2
3
2
= 6x − 6x = 0 ( , ψ(x; y) = x − 3xy (
-
(
ψ(x; y) = x3 − 3xy 2 !
)
' "(
" .
(
/ #" //%
0
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1 (
f (z)
a !
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f (z) = f (a) +
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f (a)
f (a)
(z − a) +
(z − a)2 + . . .
1!
2!
3 (1
ez = 1 + z +
sin z = z −
z2 z3
+
+ ...
2!
3!
z3 z5
+
−...
3!
5!
# %
1( )
# /%
# 4%
cos z = 1 −
+∞
Cn (z − a)n
n=0
$ '
+
½
z2 z3
+
− ...
2
3
a(a − 1) 2 a(a − 1)(a − 2) 3
z +
z + ...
2!
3!
!
z2 z4
+
− ...
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z
ln(1 + z) = z −
(1 + z)a = 1 + az +
|z| < 1
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#
$ % & $
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( #
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$ $
.
n
Cn (z − a)
a :
n=0
R
z||z − a| < R
+∞
R=
)
1
Cn+1 .
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n→+∞
Cn
! /
& #
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0 # ) *
+∞
n=0
√ 1
(z − i)n n
2n
2 '
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3
|z − i| < R
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& " i 4 $"
!
½
¾
R=
1
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n + 1 · 2n
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lim
n→+∞ 2n+1 ·
n
+∞
Cn(z − a)n
n=0
!
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"
$ %
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r < |z − a| < R 0 r +∞ 0 R +∞ % # (
#
! "
! (z − a)
f (z) =
+∞
Cn (z − a)n ,
n=−∞
) '') & % #
* !"!# $ %
&¿ f (z)'
−1
Cn (z − a)n = C−1 (z − a)−1 + C−2 (z − a)−2 + . . .
+
n=−∞
+∞
$
&' Cn(z − a)n = C0 +
n=0
+C1 (z − a) + C2 (z − a)2 + . . . (
, ) f (z) r < |z − a| < R
' $
& !"!#
-
% # % %
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a
lim f (z)
z→a
a |f (z)|
z → a f (z) z → a
a
lim f (z)
z→a
f (z)
!
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! f (z) " # $
% &
' f (z) =
+∞
n=0
Cn (z − a)n
a (
! f (z) "
# $ %
& ' f (z) =
=
+∞
Cn (z − a)n
n=−k
a
! f (z)
" # $ %
& '
+∞
f (z) =
n=−∞
Cn (z − a)n
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# $ ( +
1
( &
!
z = a
f (z)
!
1
f (z)
n − 1 #
"#!& ,
-
-
! f (z) !
" z " "
' ( ( )
*
! + , )
( ( sin z( cos z !
z = 0
sin z
f (z) =
z
! " z
z2 z4
sin z
=1−
+
− ...
z
3!
5!
# $ % z = 0 $ & $
' $ ( $ !
cos z
f (z) =
)
z = 0
z
)
! " z
cos z
1
z
z3
= − +
− ...
z
z 2! 4!
#
$ % z = 0 *+,
( $ ! - ' z = 0 ,
1
z
=
!
.$ z = 0 &+
f (z)
cos z
z
/ !
0
cos z
z cos z + z sin z
=
cos z
cos2 z
z = 0
f (x) = e1/z
1
z = 0
1
1
1
% e1/z = 1 + +
+
+...
2
z 2!z
3!z 3
2 ' & ' ' ' +
' z = 0 ! f (z) = e1/z /
& $
z = arcsin(4x2 + y 2 )
4x2 − 36y − 3z 2 = 0
√
√
x x−x+ y
z =
y
∂z
dz
∂x
dx
z = xy y = ctg x
y y
ln y + x = ln x
z = √x + 2y
M(4; 2) N (5; 3) grad z(M)
! " z = x2 + xy +
2
y
+ +x−2y # $ m $ M
2
x 0 y ≥ 0 y 10+x
M0 (1; 1; 3/2) %
"
⇒
|4x2 + y 2 | 1 ⇒ −1 4x2 + y 2 1 ⇒
4x2 + y 2 ≥ −1,
2
2
4x + y 1,
,
z2
x2
−
⇒
9
12
Ox
4x2 − 36y − 3z 2 = 0 ⇔ y =
1
a = ;b = 1
2
y
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11111
1
1/2
-1/2
x
-1
√
√
x x−x+ y
x3/2 − x
⇔z=
+ y −1/2 ⇒
y
y
√
3 1/2
x −1
3 x − 2 ∂z
x3/2 − x 1 − 3
∂z
= 2
=
=−
− y 2
⇒
∂x
y
2y
∂y
y2
2
3 1/2
x −1
∂ 2z
∂ 2z
3
2(x3/2 − x) 3 −5/2
4
= √
=
=
+ y
∂x2
y
y3
4
4 xy ∂y 2
√
3 1/2
x
−
1
2
−
3
∂ 2z
x
= −2
=
∂x∂y
y2
2y 2
z=
∂z
∂z
= y · xy−1 ;
= xy ln x ⇒
∂x
∂y
dz
∂z ∂z dy
1
=
⇒
=
+
= y · xy−1 + xy ln x − 2
dx
∂x ∂y dx
sin x
xctg · ln ctg x
= ctg x · xctg x−1 −
.
sin2 x
z = xy ; y = ctg x ⇒
1
y
y
ln y + x = ln x ⇔ + 1 = ⇒ y = − y = y
y
x
x
1
−1
x
y =
y
=y
1
1
−1 −y 2 =
x
x
x
2
1
y
2
−1 − 2 = y 1−
.
x
x
x
1
−1
x
= y
√
x + 2y M(4; 2) N (5; 3) ⇒
√
1
1
MN = (1; 1) ⇒ |MN | = 2 ⇒ cos α = √ ; cos β = √ ⇒
2
2
∂z
∂z
∂z
=
cos α +
cos β =
⇒
∂l M
∂x M
∂y M
1
1
1
2
√
√ + √
√
=
=
2 x + 2y 2 2 x + 2y 2 x = 4
y=2
z=
1 1
3
1 1
= √ √ +√ √ = ,
8
2 8 2
8 2
∂z ∂z
1
1
grad z(M) =
i
+
j = √ i + √ j.
∂x M
∂y M
2 8
8
y2
+ x − 2y ⇒
2
∂z
∂z
∂ 2z
∂ 2z
= 2x + y + 1;
= x + y − 2;
=
2;
= 1;
∂x
∂y
∂x2
∂y 2
⎧
∂z
⎪
⎨
= 0,
∂ 2z
x = −3,
2x + y + 1 = 0,
∂x
⇔
=1⇒
⇔
∂z
y
= 5.
x
+
y
−
2
=
0
⎪
∂x∂y
⎩
=0
∂y
2
∂ 2z ∂ 2z
∂ 2z
∂ 2z
Δ=
−
= 2 − 1 > 0; 2 = 2 > 0 ⇒
2
2
∂x ∂y
∂x∂y
∂x
x = −3; y = 5; z = −6,5.
x 10 + x x 0 y ≥ 0 !
" # $%& x = 0 y = 0
" # ABC #
z = x2 + xy +
y = x + 10 M(−3; 5)
z = −6,5
ABC
B(0;10)
K(-3,8;6,2)
M(-3;5)
D(0;2)
A(-10;0)
E(-0,5;0)
C(0;0)
! "
y2
dz
− 2y # y ∈ [0; 10]
= y − 2 = 0 ⇒ y = 2
• x = 0" z =
2
dy
$ m M
B(0; 10)
C(0; 0) D(0; 2)
dz
1
= 2x+1 = 0 ⇒ x = −
• y = 0" z = x2 +x # x ∈ [−10; 0]
dx
2
! m M %
1
%& ' A(−10; 0) E(− ; 0)
2
(x + 10)2
+ x − 2(x + 10) =
• y = x + 10" z = x2 + x(x + 10) +
2
dz
5 2
= x + 19x + 30(
= 5x + 19 = 0 ⇒ x = −3, 8 m M
2
dx
' K(−3, 8; 6, 2)
z(A) = z(−10; 0) = 90 z(B) = z(0; 10) = 30
z(C) = z(0; 0) = 0 z(D) = z(0; 2) = −2 z(E) = z(−0, 5; 0) = −0, 25
z(K) = z(−3, 8; 6, 2) = −6 z(M) = z(−3; 5) = −6, 5 %! )
!& z(A) = z(−10; 0) = M = 90 & z(M) = z(−3; 5) =
= m = −6, 5
*
z − z0 = fx (x0 ; y0 ) · (x − x0 ) + fy (x0 ; y0 )(y − y0 )
x0 = y0 = 1 z0 = 3/2 fx (x0 ; y0 ) = 2x0 + y0 + 1 = 4
fy (x0 ; y0 ) = x0 + y0 − 2 = 0
z − 3/2 = 4(x − 1)
8x − 2z − 5 = 0
z = ln(x2 − y 2 − 4)
4x − 2y2 − z2 = 0
x
z = arcsin y
dz
dx
z = u −1 v u = sin x v = √x
2
dy d y
dx
dx2
2y = x + ln x
!"
z =
M(3; 4)
# $ " z = x2 + y2 +
+y −2x m M 0 x 2 0 y 3
M0(1; 0; −1)
"% $
x2 + y 2
!" ! " # " $
% &
' & &
( ) &
*& & & ) " + )
) ! !,) $ (
!"
- & & +
, ) . !" f (x)
[a; b] ! , !", F (x) !
+
!" f (x)
F (x) = f (x).
/0123
4 & ,* & ! 5 ).
!" f (x) ! , !", F (x) " !
f (x)dx
dF (x) = F (x)dx = f (x) dx.
/0163
012 F (x) [a; b]
f (x)
f (x) [a; b]
%) ) F (x) [a; b]
# ! &
!" f (x) = cos 2x
!" F (x) = 12 sin 2x ! ! !
1
sin 2x = cos 2x
2
d
1
sin 2x = cos 2x dx.
2
% ! !"
, ) + ) 5 & ! .
!
"# $ % # &
'
() ( (
) * )
( (
+ ( , '
- #
. f (x) [a; b]
F (x)
F (x) + C C ! "
/ F (x) 0
f (x) F (x) + C 0 * (F (x) = f (x) ⇒ (F (x) + C) = f (x)) / '
[a; b]
f (x) ( '
( F (x) Φ(x) + % !& ,
Φ (x) = f (x),
F (x) = f (x).
1( ( '
() )
# )
,
(Φ(x) − F (x)) = 0.
% &
2 ( 3 * *
*
'
# * "#
( % &
Φ(x) − F (x) = C
Φ(x) = F (x) + C.
% &
1 % & C 0 4 (*
F (x) + C )( ) ()
'
2 (
cos 2x
1
x3
sin 2x + C, (
+ C. '
x2
2
3
( C * ( '
* ( ( - (
x1 ln x + C1 x > 0 ln(−x) + C2 x < 0
C1 C2
! x1 "
ln |x| + C #
C x > 0 x < 0
$
%&' F (x)
f (x) F (x) + C C
!
( #
$
#
f (x) dx )
f (x) dx = F (x) + C.
*%&+,
- $
,
f (x)
− ,
f (x) dx
−
.
/
$
−
cos 2x dx =
$
x2 dx =
1
sin 2x + C,
2
x3
+ C.
3
0# *
!, "
- 1# 2 "
1 2
( ! 3 #
# f (x) F (x) "
# #
!
! " !# $ % & "
' # #
$
$
sin x
dx −
x
2
e−x dx −
! ,
!.
(
) *#
$
+ # % ! & "
, !- % ! &
" !
f (x) dx,
k = 0.
. ! k
$
$
kf (x) dx = k
/
,0-
1 2 $! ! &
" ! $ % & " !/
$
$
(f (x) ± φ(x)) dx =
$
f (x) dx ±
φ(x) dx.
,3-
& ,0- ,3-
&&" ! 1 !
, 4 -
. $
! & " /
$
f (x) dx = f (x).
,5 + &&" $
/
$
d
f (x) dx = f (x) dx.
,6 !
$
dx
= d arcsin x,
1 − x2
dx
= d ln x.
x
√
!
" # ! $
! $ !
%
! & $
! ' '
( ! $ # sinx x dx e−x dx
' ) * $
$ sinx x e−x + ,
2
2
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/ 0 !. $
* ) *
0 $ $ 0 *
* $
" '
$ *
* ! *
!*
1 !
*
$ * n+1
2 n+1
$
xn dx =
1.
d
! $
x
x
=
+ C, (n = −1),
n+1 $ n+1
n=0
dx = x + C,
$
dx
= d(ln |x|) = ln |x| + C,
$ x
$
3. sin x dx = d(− cos x) = − cos x + C,
$
2.
$
4.
$
cos x dx = d(sin x) = sin x + C,
$
dx
=
d(tg x) = tg x + C,
5.
2
$ cos x $
dx
6.
= d(− ctg x) = − ctg x + C,
2
sin
x
$
$
x
1
x
1
dx
arctg
= arctg + C,
=
d
7.
2 + x2
a
a
a
a
a
$
$
dx
x
x
√
8.
= arcsin + C,
= d arcsin
a
a2 − x$2
$
$ a
x
x
a
a
x
9.
a dx = d
=
+ C,
ex dx = ex + C,
ln a
ln a
$
√
dx
√
= ln |x + x2 ± a2 | + C,
10.
x2 ± a2
$
$
x − a
a + x
1
1
dx
dx
+ C,
+ C.
ln
ln
=
=
11.
x2 − a2
2a x + a
a2 − x2
2a a − x
$
!
" # # $ % #
& # % & &" % ' %
& ( & % ) # #&
& % &
*+ ,
x
x
*+ +
! " n < 0# $ % & !' !!
( C )
( )
% - %
# -
) # # #
%
. # ) "
% # &
$
$
! $" #
f (x)dx =
d
f (φ(t))φ (t)dt,
f (x)dx = f (x)dx,
$
d
f (φ(t))φ(t)dt = f (φ(t))d(φ(t)) = |x = φ(t)| = f (x)dx.
$ " % "
&% '( x = φ(t) t : t = ψ(x)
% Φ(t) :
$
$
f (x)dx =
f (φ(t) · φ (t))dt = Φ(t) + C = Φ(ψ(x)) + C.
#
f (x)dx
x φ(t) dx φ(t)dt
)
!"
# "
ϕ(x) = t
#
$%
I = x2ex dx.
* ( # + % % ,
- x3 x2. . % x
( ' +/,
' x3 = t, ,
% " 3x2dx = dt. .
x2 01 dt.
2 % -3 3,
- '#
3
1
I = x3 = t ⇒ 3x2 dx = dt =
3
4 $%
$
1
1 3
et dt = et + C = ex + C.
3
3
I =
#
sin kx dx.
dt
k
$
1
1
1
sin tdt = − cos t + C = − cos kx + C .
I=
k
k
k
kx = t =⇒ dx =
$
I=
√
t=x+
√
dx
.
± a2
x2
x2 ± a2 ⇒
√ 2
x ± a2 + x
x
= √
dx ⇒
⇒ dt = 1 + √
2
2
x ±a
x2 ± a2
dt
dx
dt
dx
√
=
⇒√
= .
⇒√
t
x2 ± a2
x + x2 ± a2
x2 ± a2
$
I=
√
dt
= ln |t| + C = ln |x + x2 ± a2 | + C.
t
! " # $
% & $ ' " #
'( #" " ) *+
$ $
I = # (x − 3)2dx
$
I=
$
2
(x − 6x + 9)dx =
2
x dx − 6
$
$
xdx + 9
dx =
x3
x2
x3
+ C1 ) − (6 + C2 ) + (9x + C3 ) =
− 3x2 + 9x + C.
3
2
3
, -
*( ' ( . ( C1 − C2 + C3
' . . .- / & ' C
=(
+
( & $ '( $ (
' ' ' $
0 I =
$
x2 − 3
√ dx.
x
$
3
2
(x − 3x
I=
− 21
3
1
√
x− 2 +1
2 √
x 2 +1
)dx = 3
−3 1
+ C = x2 x − 6 x + C.
5
+
1
−
+
1
2
2
I =
$
I=
sin2 x + cos2 x
dx =
sin2 x cos2 x
$
$
dx
sin2 x cos2 x
dx
+
cos2 x
$
dx
= tg x − ctg x + C.
sin2 x
$
dx
=2
sin2 2x
I=4
$
$
d(2x)
= −2 ctg 2x + C.
sin2 2x
#
I = tg5 x dx.
$
1
dx
− 1 dx = tg 3 x 2 −
I = tg x tg xdx = tg x
cos2 x
cos x
$
$
$
$
tg 4 x
− tg 3 xdx.
− tg 3 xdx = tg 3 xd tg x − tg 3 xdx =
4
n = 5 n = 3
3
$
2
3
$
1
tg4 x
− tg x
−
1
dx =
4
cos2 x
$
$
tg4 x
tg4 x tg2 x
=
− tg xd tg x + tg xdx =
−
− ln | cos x| + C.
4
4
2
I=
!
1
(sin(n − m)x + sin(n + m)x) ,
2
1
sin nx sin mx = (cos(n − m)x − cos(n + m)x) ,
2
1
cos nx cos mx = (cos(n − m)x + cos(n + m)x) .
2
sin nx cos mx =
"
#$% &'(
$
I=
#
sin 2x cos 3x dx.
1
1
1
(− sin x + sin 5x)dx = cos x −
cos 5x + C.
2
2
10
I=
$
I=
dx
.
x2 − a2
1
1
=
x2 − a2
2a
1
1
−
x−a x+a
.
!"
!# $ % !"
$
$
d(x − a)
d(x + a)
1
−
=
I=
2a
x−a
x+a
x − a
1
1
+ C.
=
(ln |x − a| − ln |x + a|) =
ln
2a
2a x + a
& ' ( ) * ++
, d(uv) = udv + vdu- ) % )
" !
$
$
udv = uv − vdu.
. +/
0 !" . +/ ( 1 !" ) !
, ! )
2/ () 1"3 4
"# ) ! "# 1"3
$
$
$
Pn (x)eαx dx
Pn (x) sin αx dx
Pn (x) cos αx dx.
$
Pn (x)eαx dx =⇒
eαx
dv = e dx = d
α
Pn (x)dx,
αx
u = Pn (x), du =
$
Pn (x) sin αx dx =⇒
, v=
eαx
.
α
cos αx
cos αx
, v=−
.
u = Pn (x), du = Pn (x)dx, dv = sin αx dx = d −
α
α
$
Pn (x) cos αx dx =⇒
sin αx
sin αx
, v=
.
u = Pn (x), du = Pn (x)dx, dv = cos αx dx = d
α
α
!
" " "
$
Pn (x) ln x dx
$
Pn (x) arcsin αx dx
$
Pn (x) arctg αx dx.
$
Pn (x) ln x dx =⇒
dx
, dv = Pn (x)dx, v =
u = ln x, du =
x
$
Pn (x) arcsin αx dx =⇒
u = arcsin αx, du =
$
αdx
1 − (αx)2
$
Pn (x) dx,
$
, dv = Pn (x)dx, v =
Pn (x) dx,
Pn (x) arctg αx dx =⇒
u = arctg αx, du =
αdx
, dv = Pn (x)dx, v =
1 + (αx)2
$
#
$%&'( I = # x2 ln x dx.
Pn (x) dx.
u = ln x
du = dx
x =
I =
3
3
x
x
2
dv = x dx = d 3 v = 3
$
x3
1
x3
x3
=
ln x −
x2 dx =
ln x −
+ C.
3
3
3
9
#
I = x arctg x dx.
$
u = arctg x
1
du = x2dx+1 x2
x2 dx
arctg
x
−
=
=
I =
2
2
x
x
dv = xdx = d 2 v = 2
2
2
x2 + 1
$ 2
x2
1
x +1−1
x2
=
arctg x −
dx =
arctg x−
2
2
2
x +1
2
$
$
1
1
dx
x2
1
1
−
dx +
=
arctg x − x + arctg x + C.
2
2
2
x +1
2
2
2
# 2
I = x sin x dx.
u = x2
du = 2xdx
I =
dv = sin xdx = d(− cos x) v = − cos x
$
u=x
+ 2 x cos xdx =
dv = cos xdx = d(sin x)
= −x2 cos x + 2x sin x + 2 cos x + C.
I=
#
= −x2 cos x+
du = dx
=
v = sin x
xe3x dx.
$
u=x
du = dx 1 3x 1
e3x dx =
I =
−
1 3x = xe
3x
dv = e dx v = 3 e
3
3
1
1
= xe3x − e3x + C.
3
9
! """
# $%& & & ' (
$ )
* ! &
I = # eax cos nx dx.
u = eax
du = aeax dx
=
I =
1
dv = cos nx dx v = n sin nx
$
1
a
= eax sin nx −
eax sin nx dx =
n
n
u = eax
du = aeax dx 1 ax
=
= e sin nx−
dv = sin nx dx v = − n1 cos nx n
$
1
a
a
− eax cos nx +
eax cos nx dx =
−
n
n
n
a ax
a2
1 ax
= e sin nx + 2 e cos nx − 2 I.
n
n
n
I
$
I=
eax cos nx dx =
eax (a cos nx + n sin nx)
+ C.
a2 + n2
C !
"# $ "
√
%&'( I = # x2 − a2 dx.
√
$
√
u = x2 − a2 du = √ xdx
x2 dx
x2 −a2 = x x2 − a2 −
√
=
I =
dv = dx
v=x
x2 − a2
$
$
√
√
√
x2 − a2 + a2
√
= x x2 − a2 −
dx = x x2 − a2 −
x2 − a2 dx−
x2 − a2
$
√
√
dx
= x2 x2 − a2 − I − a2 ln |x + x2 − a2 | + C1 .
− a2 √
2
2
x −a
)
I=x
2
√
x2
−
a2
− I − a2 ln |x +
√
x2 − a2 | + C1 ,
√
1 √
a2
I = x x2 − a2 − ln |x + x2 − a2 | + C.
2
2
!
"
f (x) = cos12 2x .
# $ % F (x) = 12 tg 2x + C.
& "% f (x) = F (x) = 12 tg 2x = 21 · 2 cos12 2x = cos12 2x .
' f (x) = x4.
x5
+ C.
5
x5
1
+ C = · 5x4 + 0 = x4 .
f (x) = F (x) =
5
5
# $ %
& "%
F (x) =
f (x) = x2 .
# $ % F (x) = 2 ln | − x| + C.
& "% f (x) = F (x) = (2 ln | − x| + C) = 2(− x1 )(−1) + 0 = x2 .
'
( )
!
& *
(! + !
" ) ! ,
" "
# " " -
(!. / "
0 ! ! 1 !
%
I = # sin3 x cos x dx.
# $ %
2/!) / d(sin x) = cos xdx,
$
I=
sin3 xd sin x =
sin4 x
+ C.
4
$
I=
$ √
arctg x
dx.
1 + x2
I =
(arctg x)1/2 d arctg x =
d(arctg x) =
(arctg x)1/2+1
1/2 + 1
dx
,
1 + x2
2
= (arctg x)3/2 + C.
3
! " # $ %
& I1 =
#
tg x dx, I2 =
#
ctg x dx.
$
$
d cos x
sin x dx
=−
= − ln | cos x| + C.
I1 =
cos x
cos x
$
$
d sin x
cos x dx
=
= ln | sin x| + C.
I2 =
sin x
sin x
$
dx
.
' I =
x ln x
I=
$
dx
=
x ln x
$
dx/x
=
ln x
$
d ln |x|
= ln | ln |x|| + C.
ln x
!
#
( I = sin 5x dx.
!
# $
) * %
+ * $ +
15 :
$
1
1
sin 5x d5x = − cos 5x + C.
I=
5
5
#
, I = ex cos ex dx.
-* ** d(ex ) = ex dx,
$
I = cos ex dex = sin ex + C.
!"#$ I =
$
% &
dx
.
x2 cos2 x1
$
I=−
!"## I =
% &
I=
1
3
!"#'
% &
1
I=
4
$
!"#!
$
d x1
1
= tg + C.
x
cos2 x1
dx
.
sin2 (3x − 5)
$
d(3x − 5)
1
= − ctg(3x − 5) + C.
3
sin2 (3x − 5)
$
3
x dx
I=
.
5 − x8
√
5 + x4
dx4
1
√
= √ ln √
+ C.
( 5)2 − (x4 )2
8 5 5 − x4
#
2
I = x3x dx.
% &
1
I=
2
$
2
3x dx2 =
2
3x
+ C.
2 ln 3
%
( ) *)
&
$
$
$
(f (x) ± ϕ(x)) dx = f (x) dx ± ϕ(x) dx,
$
$
kf (x) dx = k f (x) dx.
I=
$
x4 − 10x2 + 5
dx.
x2
$
$
$
$
5
I=
x2 − 10 + 2 dx = x2 dx − 10 dx + 5 x−2 dx =
x
5
x3
− 10x − + C.
3
x
# √
2
dx.
I=
2x +
x
=
√ $ 1
√ $
√
1
I = 2 x 2 dx + 2 x− 2 dx = 2
#
I = cos 2x cos 5x dx.
√
2 √
x x + 2 x + C.
3
$
$
1
1
(cos 7x + cos 3x) dx =
cos 7xd(7x)+
I=
2
14
$
1
1
1
+
cos 3x d(3x) =
sin 7x + sin 3x + C.
6
14
6
! "# " " $ # "#
# "%& '"# "
$
√
3
( I = 9x2 x3 + 10 dx.
3
1
+1
x + 10 = t
#
3
= 3 t 13 dt = 3 t
I = 2
+C =
1
3x dx = dt
+1
3
√
√
3
= 94 t t + C = 94 (x3 + 10)3 x3 + 10 + C.
$
4xdx
√
) I =
.
5
8 − x2
− 15 +1
8 − x2 = t
#
= −2 t− 15 dt = −2 t
+C =
I =
−2x dx = dt
−1 + 1
4
4
= − 25 t 5 + C = − 52 (8 − x2 ) 5 + C.
5
2 cos x dx
.
4 + sin x
4 + sin x = t
I =
cos x dx = dt
I =
= 2 ln(4 + sin x) + C.
$
$
dt
= 2
= 2 ln |t| + C =
t
4 + sin x > 0
I=
$
arcsin x
dx.
1 − x2
arcsin x = t
√
#√
=
I = √ dx
t dt = 23 t t + C =
=
dt
1−x2
√
= 32 arcsin x arcsin x + C.
! "#
√
1 − x2 $ "
#
$ √
$ √
dx
I=
arcsin x √
=
arcsin x d(arcsin x) =
1 − x2
√
2
= arcsin x arcsin x + C.
3
%
&
#
#
I = ex (ex + 2)2 dx.
! "# $ "
'
#
(ex + 2)3
+ C.
I = (ex + 2)2 d(ex + 2) =
3
$ x
e + sin x
dx.
I=
ex − cos x
$
d(ex − cos x)
= ln |ex − cos x| + C.
I=
ex − cos x
$
x+5
√
dx.
I =
x2 + 3
$
$
dx
d(x2 + 3)
1
x dx
√
+5 √
=
+
I = √
2
x2 + 3
x2 + 3
x2 + 3
$
√
√
dx
= x2 + 3 + ln |x + x2 + 3| + C.
2
(x + 3)
I=
$
x4 dx
.
1 + x10
$
dx5
1
1
= arctg x5 + C.
5
2
5
1 + (x )
5
#
I = xeax dx.
$
u=x
du = dx xeax 1
−
eax dx =
I=
1 ax =
ax
dv = e dx v = a e
a
a
xeax eax
− 2 + C.
=
a
a
#
I = xn ln x dx (n = −1).
u = ln x
#
xn+1
du = dx
x
ln x+ xn−1 dx =
=
I =
n+1
x
n
dv = x dx v = n+1
n
+
1
1
xn+1
ln x −
+ C.
=
n+1
n+1
! " n = −1, #$ !
$
$
1
dx
= ln x d(ln x) = ln2 x + C.
I = ln x
x
2
# ax
% I = e sin nx dx.
I=
u = eax ,
1
du = aeax dx
= − eax cos nx+
I=
dv = sin nx dx, v = − n1 cos nx
n
$
u = eax
a
du = aeax dx
=
eax cos nx dx =
+
dv = cos nx dx v = n1 sin nx
n
$
a 1 ax
a
1
e sin nx −
eax sin nx dx =
= − eax cos nx +
n
n n
n
a
a2
1 ax
= − e cos nx + 2 − 2 I. ⇒
n $
n
n
eax (a sin nx − n cos nx)
ax
+ C.
⇒ I = e sin nx dx =
a2 + n2
# & !'
eax cos nx dx.
!
( #
f (x) = 6x2.
f (x) = tg 5x.
I = # √x dx 4 .
2−x
! I = # sincosx2xdx .
2
x dx
√
.
I = # cos
sin x
I = # (1 − 7x)5x dx.
I = # etg x cosdx2 x .
3
#
3
x
" I = arcsin
dx.
1 − x2
# x +2x
# I = e (x + 1) dx.
$ I = # √xx2dx− 6 .
2
I = # e
√
x
dx
√ .
x
I = # √9 − exex dx.
I = # (√x − 1)2 dx.
5
! I = # (x −x 1) dx.
I = # (sin 5x cos x) dx.
3
% &'
√
I = # x 2x2 + 7 dx.
+ 5) dx
.
I = # x(2x
2 + 5x − 13
I =
# 2 + ln x
.
x
# sin 2x dx
.
" I =
7 + cos2 x
# x + x3
dx.
# I =
x4 + 5
#
% I = x sin 2x dx.
#
& I = arctg x dx.
#
' I = e2t cos 3t dt.
! I =
(
#
sin 3x dx
√
.
5 + cos 3x
#√
dx
I = 3 tg 2x 2 .
cos 2x
$
) * + * , -
*
.
*
, * $ /*
A
,
I.
x−a
A
II.
(n = 2, 3, ...),
(x − a)n
Mx + N
III. 2
(D = p2 − 4q < 0),
x + px + q
Mx + N
IV. 2
(D = p2 − 4q < 0, n = 2, 3...).
(x + px + q)n
!
$
d(x − a)
Adx
=A
= A ln |x − a| + C.
x−a
x−a
$
$
(x − a)−n+1
Adx
+C =
= A (x − a)−n d(x − a) = A
II.
n
(x − a)
−n + 1
A
=
+ C.
(1 − n)(x − a)n−1
$
I.
"
# $ ax2 + bx + c #
"%
1
b
(ax2 + bx + c) = ax + = t.
2
2
1 2
(x + pq + q) = t dx = dt
Mx + N
dx = 2 p
III.
x=t−
x+ 2 =t
x2 + px + q
$
$
p
Mt + (N − Mp
M(t − 2 ) + N
)
2
dt =
dt.
=
p 2
p
P2
2
(t − 2 ) + p(t − 2 ) + q
t + (q − 4 )
&'' (
$
) % *
q−
p
2
=
p2
= a2 > 0,
4
$
$
dt
Mx + N
tdt
Mp
dx
=
M
+
N
−
=
x2 + px + q
t2 + a2
2
t2 + a2
t
Mp 1
M
ln(t2 + a2 ) + N −
arctg + C.
=
2
2
a
a
$
+ t a # *
$
=
Mx + N
dx =
x2 + px + q
N − Mp
x + p2
M
2
ln(x2 + px + q) +
arctg
+ C.
2
2
2
q − p4
q − p4
&''!(
! ! 2x + p. "
2x+p Mx+N. # 2x+p
N/2 N − Mp/2.
$
(2x + p)
Mp
M
+N −
= Mx + N.
2
2
%
Mx + N
x2 + px + q
(2x + p) M2 + N −
x2 + px + q
Mp
c
&
N − Mp
M 2x + p
2
+
.
2 x2 + px + q x2 + px + q
' #
( ) !
p 2
p2
+q− .
x2 + px + 1 = x +
2
4
*
+ ,,
+ - 4q − p2 > 0.
..!,
4q − p2 < 0,
ax2 + bx + c,
a
x2 + px + q
..!,
$
I=
x+1
dx
x2 + 4x + 8
t t = x + 2 x = t − 2 dx = dt
$
$
$
t−2+1
t−1
x+1
dx
=
dt
=
dt =
I=
x2 + 4x + 8
(t − 2)2 + 4(t − 2) + 8
t2 + 4
$
$
tdt
dt
1
t
1
=
−
= ln(t2 + 4) − arctg + C =
t2 + 4
t2 + 4
2
2
2
1
x+2
1
= ln(x2 + 4x + 8) − arctg
+ C.
2
2
2
! !
" " 2x + 4 #
$ %! ! & $ !' ($
x+1
1 2x + 4 − 2
=
.
x2 + 4x + 8
2 x2 + 4x + 8
' ! "
' " '
1 2x + 4 − 2
1 2x + 4
1
=
−
.
2 x2 + 4x + 8
2 x2 + 4x + 8 (x + 2)2 + 4
)
$
d(x2 + 4x + 8)
d(x + 2)
−
=
2
x + 4x + 8
(x + 2)2 + 22
x+2
1
1
+ C.
= ln(x2 + 4x + 8) − ln arctg
2
2
2
I=
1
2
$
*+ ! *+
! ***
$
Mx + N
dx =
(x2 + px + q)n
$
$
dt
Mp
tdt
+
N
−
.
=M
(t2 + a2 )n
2
(t2 + a2 )n
$
'
tdt
1
=
(t2 + a2 )n
2
$
,(( -.
,(( -. '! "!"
(t2 + a2 )−n d(t2 + a2 ) =
1
+ C.
2(1 − n)(t2 + a2 )n−1
$
In =
=
1
a2
#
In =
#
#
dt
(t2 +a2 )n
$ 2
(t + a2 ) − t2
dt
1
=
dt =
2
2
n
2
(t + a )
a
(t2 + a2 )n
$
$
dt
t2 dt
.
−
(t2 + a2 )n−1
(t2 + a2 )n
dt
(t2 +a2 )n−1
In =
t2 dt
(t2 +a2 )n
= In−1
1
a2
In−1 −
$
t2 dt
2
(t + a2 )n
.
u=t
du = dt
1
dv = 2 tdt2 n v =
(t +a )
2(1−n)(t2 +a2 )n−1
$
t2 dt
t
1
In−1 .
=
−
2
(t + a2 )n
2(1 − n)(t2 + a2 )n−1 2(1 − n)
! "
2n − 3
t
In−1 +
2n − 2
2(n − 1)(t2 + a2 )n−1
#
$ % &
% ' (
! )* & n &
! )* & n − 1
+ # , n − 1 & & ,
-
. I3 = # (t2 +dt 1)3
In =
/
, a
1
I3 = 2
1
I2 =
$
1
a2
0
= 1, n = 3
2·3−3
t
I2 +
2·3−2
2(3 − 1)(t2 + 1)2
.
#
3
t
= I2 +
.
4
4(t2 + 1)2
1
dt
2·2−3
t
t
I1 +
= I1 +
.
=
(t2 + 1)2
2·2−2
2(2 − 1)(t2 + 1)
2
2(t2 + 1)
$
I1 =
I2 =
dt
= arctg t + C,
t2 + 1
t
1
arctg t +
+C
2
2(t2 + 1)
1
t
t
arctg t +
+
+C =
2
2(t2 + 1)
4(t2 + 1)2
3
3t
t
+ arctg t + C.
+
=
2
2
2
4(t + 1)
8(t + 1) 8
3
I3 =
4
#
R(x)dx R(x)
# ! " #$ % &
R(x)dx
" #$
$
I=
2x2 + 5x − 8
dx.
(x − 1)3 (x + 2)2
D
A
B
C
E
2x2 + 5x − 8
+D
.
=
+
+
+
(x − 1)3 (x + 2)2
(x − 1)3 (x − 1)2 x − 1
(x + 2)2 x + 2
! "
2x2 + 5x − 8 = A(x + 2)2 + B(x − 1)(x + 2)2 + C(x − 1)2 (x + 2)2 +
+ D(x − 1)3 + E(x − 1)3 (x + 2).
#$%%& ' A, B, C, D, E ! (
! ') ! * ') $%%&
⎧
x = 1 ⇒ −1 = 9A ⇒ A = − 19 ,
⎪
⎪
⎪
10
⎪
⎪
⎨ x = −2 ⇒ −10 = −27D ⇒ D = 27 ,
x = 2 ⇒ 10 = 16A + 16C + D + 4E ⇒ 4B + 4C + E = 77
,
27
⎪
⎪
4
⎪
⇒
C
+
E
=
0,
x
⎪
⎪
⎩
⇒ −8 = 4A − 4B + 4C − D − 2E ⇒ 2B − 2C + E =
B, C, E
⎧
⎪
⎨C + E = 0,
2B − 2C + E =
⎪
⎩4B + 4C + E =
!
97
.
27
97
,
27
77
,
27
"
B=
13
13
29
, C=− , E= .
27
27
27
# $ !
$
$
2x2 + 5x − 8
−1/9
29/27
−13/27
+
dx
=
(
+
+
(x − 1)3 (x + 2)2
(x − 1)3 (x − 1)2
x−1
10/27
1
1
13/27
29 1
+
)dx =
−
+
−
(x + 2)2
x+2
18 (x − 1)2 27 x − 1
10 1
13
13
ln |x − 1| −
+
ln |x + 2| + C =
−
27
27 x + 2 27
13 x + 2
26x2 + 5x − 34
+
ln
+ C.
=−
18(x − 1)2 (x + 2) 27 x − 1
%%%
$
I=
x4 + 5x3 − 6x + 5
dx.
x3 + 2x2 − 1
&
" ' $! ( ) !
* $"
x4 + 5x3 − 6x + 5
−6x2 − 5x + 8
= x+3+ 3
.
3
2
x + 2x − 1
x + 2x2 − 1
x3 + 2x2 − 1
x = −1
x + 1
x3 + 2x2 − 1 = (x + 1)(x2 + x − 1).
! "# x2 + x − 1
$
%
&
−6x2 − 5x + 8
A
Bx + C
=
+
.
(x + 1)(x2 + x − 1)
x + 1 x2 + x − 1
' &
−6x2 − 5x + 8 = A(x2 + x − 1) + (Bx + C)(x + 1) =
= (A + B)x2 + (A + B + C)x − A + C.
!(( % A, B, C
)
# " # (( %
⎧
⎪
⎨ x = −1 =⇒ 7 = −A =⇒ A = −7,
x2 ⇒ −6 = A + B =⇒ B = 1,
⎪
⎩ ⇒ 8 = −A + C =⇒ C = 1.
*
−6x2 − 5x + 8
−7
x+1
=
+
.
(x + 1)(x2 + x − 1)
x + 1 x2 + x − 1
&
)
x+1
1 2x + 1 + 1
1 2x + 1
1
1
= · 2
=
+
.
x2 + x − 1
2 x +x−1
2 x2 + x − 1 2 (x + 12 )2 − 54
1
2x + 1
1
1
7
+ ·
+
I=
dx =
x+3−
x + 1 2 x2 + x − 1 2 (x + 12 )2 − 54
$
$
$
$
d(x + 12 )
d(x2 + x − 1) 1
1
+
=
= x dx − 3
dx +
2
x2 + x − 1
2
(x + 12 )2 − 54
√
1
1 x + 12 − 25
x2
√ + C.
+ 3x − 7 ln |x + 1| + ln |x2 + x − 1| + ln
=
2
2
2 x + 1 + 5
$
2
$
I=
4
2
3
x + 5x − 7x2 + 5
dx.
x3 − x2 + 5x − 5
!
x4 + 5x3 − 7x2 + 5
6x2 + 25x − 35
=
x
+
6
−
.
x3 − x2 + 5x − 5
x3 − x2 + 5x − 5
" "
3
x − x2 + 5x − 5 = x2 (x − 1) + 5(x − 1) = (x − 1)(x2 + 5).
" # !
A
Bx + C
6x2 + 25x − 35
=
+ 2
.
x3 − x2 + 5x − 5
x−1
x +5
$ !
#
6x2 + 25x − 35 = A(x2 + 5) + (Bx + C)(x − 1).
%&''( ) A, B, C ! )*
! +)* &''(
⎧
2
⎪
⎨ x = 1 ⇒ −4 = 6A =⇒ A = − 3 ,
2
x ⇒ 6 = A + B =⇒ B = 6 − A = 6 +
⎪
⎩ ⇒ −35 = 5A − C =⇒ C = 95 .
3
2
3
=
20
,
3
6x2 + 25x − 35
2
1
20x/3 + 95/3
=− ·
+
.
3
2
x − x + 5x − 5
3 x−1
x2 + 5
$
I=
x2
2
x2
=
2
=
1
20x/3 + 95/3
2
+
=
x+6+ − ·
3 x−1
x2 + 5
$
$
2
20
x dx
95
dx
+ 6x − ln |x − 1| +
+
=
2
2
3
3
x +5
3
x +5
x
1 (x2 + 5)10
95
+ 6x + ln
+ √ arctg √ + C.
3
(x − 1)2
3 5
5
! "
#
$ #
$
.
%%#& I = x dx
−5
' (
$
I=
d(x − 5)
= ln |x − 5| + C.
x−5
%%#)
' ($
I=
(x + 2)−4 d(x + 2) =
I=
x+3
dx.
x2 + 4x + 29
' (
$
dx
.
(x + 2)4
1
(x + 2)−4+1
+C =−
+ C.
−4 + 1
3(x + 2)3
%%#*
$
I =
$
1
t+1
dt =
I = | (x2 + 4x + 29) = x + 2 = t, dx = dt| =
2
t2 + 25
$
$
dt
1
t
1
t dt
+
= ln(t2 + 25) + arctg + C =
=
t2 + 25
t2 + 25
2
5
5
x+2
1
1
+ C.
= ln(x2 + 4x + 29) + arctg
2
5
5
I4 =
$
dx
.
(x2 + 1)4
$
dx
1 2n − 3
x
I
In =
=
+
;
n−1
(x2 + a2 )n
a2 2n − 2
2(n − 1)(x2 + a2 )n−1
5
x
;
I4 = I3 +
6
6(x2 + 1)3
3
x
;
I3 = I2 +
4
4(x2 + 1)2
1
x
.
I2 = I1 +
2
2(x2 + 1)
$
dx
= arctg x + C
I1 =
2
x +1
1
2
arctg x +
;
3
2(x2 + 1)
1
1
3 2
arctg x +
+
=
I3 =
2
2
4 3
2(x + 1)
4(x + 1)2
1
3
1
= arctg x +
+
;
2
8(x2 + 1) 4(x2 + 1)2
3
1
5 1
1
I4 =
arctg x +
+
=
+
4 2
8(x2 + 1) 4(x2 + 1)2
6(x2 + 1)3
5
15
5
1
= arctg x +
+
+
+ C.
2
2
2
2
8
32(x + 1) 16(x + 1)
6(x + 1)3
I2 =
I =
$
x4 − 3x3 − 5x2 + 30x − 22
dx.
x3 − x2 − 8x + 12
!
" # # $
x4 − 3x3 − 5x2 + 30x − 22
x2 + 2x + 2
=x−2+ 3
.
3
2
x − x − 8x + 12
x − x2 − 8x + 12
$
I=
=
x2 + 2x + 2
3
x − x2 − 8x + 12
x−2+
x2
− 2x +
2
$
x3
dx =
x2 + 2x + 2
dx.
− x2 − 8x + 12
x3 − x2 − 8x + 12 = (x − 2)2(x + 3)
x3
x2 + 2x + 2
x2 + 2x + 2
A
B
C
=
=
+
.
+
2
− x − 8x + 12
(x − 2)2 (x + 3)
x − 2 (x − 2)2 x + 3
x2 + 2x + 2 = A(x − 2)(x + 3) + B(x + 3) + C(x − 2)2 .
!"" A B C
# $ # "" %
I=
⎧
⎪
⎨ x = 2 ⇒ 10 = 5B ⇒ B = 2,
x = −3 ⇒ 5 = 25C ⇒ C = 15 ,
⎪
⎩ x2 ⇒ 1 = A + C ⇒ A = 4 .
5
4
2
1
x2
− 2x + ln |x − 2| −
+ ln |x + 3| + C.
2
5
x−2 5
&&'( I =
$
x2 − 5x + 9
dx.
(x − 1)2 (x2 + 2x + 2)
) % * +
) $ %
'
x2 − 5x + 9
A
=
+
(x − 1)2 (x2 + 2x + 2)
x−1
+
Cx + D
B
.
+
(x − 1)2 x2 + 2x + 2
x2 − 5x + 9 = A(x − 1)(x2 + 2x + 2) + B(x2 + 2x + 2)+
+ (Cx + D)(x − 1)2 ,
x2 − 5x + 9 = (A + C)x3 +
+ (A + B − 2C + D)x2 + (2B + C − 2D)x + (−2A + 2B + D).
x
⎧
x3
⎪
⎪
⎨ 2
x
x
⎪
⎪
⎩ :
⇒
⇒
⇒
⇒
A + C = 0,
A + B − 2C + D = 1,
2B + C − 2D = −5,
−2A + 2B + D = 9.
7
21
7
A = − , B = 1, C = , D = .
5
5
5
!
$
x2 − 5x + 9
dx =
(x − 1)2 (x2 + 2x + 2)
$
$
$
7
dx
dx
x+3
7
=−
+
dx.
+
5
x−1
(x − 1)2 5
x2 + 2x + 2
I=
"
#!$
1 2
(x + 2x + 2) = t ⇒ t = x + 1, x = t − 1; dx = dt.
2
% "
$
$
$
$
x+3
t+2
t dt
dt
dx
=
dt
=
+
2
=
x2 + 2x + 2
t2 + 1
t2 + 1
t2 + 1
1
1
= ln(t2 + 1) + 2 arctg t + C = ln(x2 + 2x + 2)+
2
2
+ 2 arctg(x + 1) + C.
% &
1
7
7
14
+ (x2 + 2x + 2) +
arctg(x + 1) + C.
I = − ln |x − 1| −
5
x − 1 10
5
''( I =
$
2x + 2
dx.
(x − 1)(x2 + 1)2
Bx + C
Dx + E
A
2x + 2
+ 2
+ 2
=
.
(x − 1)(x2 + 1)2
x−1
x +1
(x + 1)2
2
2x + 2 = A(x + 1)2 + (Bx + C)(x − 1)(x2 + 1) + (Dx + E)(x − 1)
2x + 2 = (A + B)x4 + (C − B)x3 + (2A + D + B + C)x2 +
+ (E − D + C − B)x + (A − C − E).
!
⎧
A + B = 0,
⎪
⎪
⎪
⎪
⎨ C − B = 0,
2A + D + B + C = 0,
⎪
⎪
E−D+C −B =2
⎪
⎪
⎩ A − C − E = 2.
"# A = 1 B = −1 C = −1 D = −2 E = 0$
%
$
2x
1
x+1
− 2
dx =
I=
−
x − 1 (x + 1)2 x2 + 1
$
$
$
$
d(x2 + 1) 1
d(x2 + 1)
dx
dx
−
−
=
−
=
2
2
2
2
x−1
(x + 1)
2
x +1
x +1
1
1
− ln(x2 + 1) − arctg x + C.
= ln |x − 1| + 2
x +1 2
& $
I=
''$(
I =
''$)
$ ''$*+
I =
2x + 5
dx.
x2 + 2x + 5
$
$
dx
.
x+3
dx
.
(x − 2)5
x+1
dx.
I=
x2 + 4x + 5
$
dx
.
I =
x3 − 2x2 + x
$
x dx
I =
.
(x − 1)(x + 1)2
$
dx
I =
.
(x3 − 1)2
$
x dx
I =
.
(5x2 + 2x + 4)2
$
!
" #
$%
& ! ' #
(
$
) *
R(sin x, cos x) dx,
R(sin x, cos x) + $%
! x2 y 3 + $% x y,
2
sin
x cos3 y + $% √ sin√
x cos y, ,
√ 2 √
( 2) ( 5 5)3 + $% 2 5 5.
-' ' ! $%
% x . '
#
% (
2 tg x2
1 − tg2 x2
sin x =
,
cos
x
=
.
1 + tg2 x2
1 + tg2 x2
/0 % ". $ % (
2t
x
1 − t2
tg = t, sin x =
, cos x =
,
2
2
1+t
1 + t2
dx =
2dt
.
1 + t2
) *
#
sinn x cosm xdx
2k + 1 (k 0, k ∈ Z)
m = 2k + 1
cos x = t
!"
$
m,
n
sin x = t
!
n = 2k + 1
#
##
2k + 1
)
I=
%& '
#
sin2 x cos3 x dx.
$
*
$
$
I = | sin x = t, cos xdx = dt| =
(1 − t2 )t2 dt =
$
(t2 − t4 )dt =
sin3 x sin5 x
t3 t5
− +C =
−
+ C.
3
5
3
5
=
* #
$
## $
$
I = sin2 x(1 − sin2 x) cos xdx = sin2 x(1 − sin2 x)d sin x =
$
sin3 x sin5 x
−
+ C.
= (sin2 x − sin4 x)d sin x =
3
5
(
)
%& %
$
+
cos x
$
I=
cos5 x
√
dx.
3
sin x
## $
cos xdx = d(sin x),
$
cos 4 x = (1 − sin2 x)2 .
##
(
"
$
1 − 2 sin 2 x + sin 4 x
d(sin x) =
I=
sin 1/3 x
$
$
−1/3
= (sin x)
d(sin x) − 2 (sin x)5/3 d(sin x)+
$
3
3
+ (sin x)11/3 d(sin x) = (sin x)2/3 − (sin x)8/3 +
2
4
3
(sin x)14/3 + C.
+
14
! "# $ % & &' (
) &
sin2 x =
sin 2x
1 − cos 2x
1 + cos 2x
, cos2 x =
, sin x cos x =
.
2
2
2
+
!!
*#
! &' ,
I=
#
sin4 x cos2 x dx
- . $
$
1 − cos 2x sin2 2x
·
dx =
sin2 x(sin x cos x)2 dx =
2
4
$
$
1
1
=
sin2 2xdx −
sin2 2x cos 2xdx.
8
8
I=
+ !/ &&
+% &
0 ! / && ,
$
$
1 − cos 4x
1
1
dx −
sin2 2xd(sin 2x) =
8
2
16
1
1
1
sin 4x −
sin3 2x + C.
= x−
16
64
48
I=
1 !/
$ % & cosk x0
(sink x)0 k = |m+n|
∈ N 0 !! )) 1 (
2
# ! 0 m n 1
1 0 !
$
$
I=
dx
.
cos3 x sin x
$
dx
tg2 x + 1
cos2 x
=
d tg x =
I=
sin x cos x
tg x
$
$
tg2 x
d tg x
=
+ ln | tg x| + C.
= tg x d tg x +
tg x
2
I=
$
dx
.
sin3 x cos5 x
dx
= d tg x,
cos2 x
3
1 + tg2 x
1
1
=
.
=
3
tg x
sin3 x cos3 x
tg x
1
√
·√
2
2
1+tg x
1+tg x
3
$
$
1
1 + 3 tg2 x + 3 tg 4 x + tg 6 x
1 + tg2 x
d tg x =
I=
d tg x =
tg x
3
tg3 x
$
3
1
tg −3 x +
+ 3 tg x + tg 3 x d tg x =
=
3
3 tg x
3
1
1
= ctg 2 x + 3 ln | tg x| + tg 2 x + tg 4 x + C.
2
2
4
! "# $%&'" %( $ &
) * !) $!
+ R(sin x, cos x) & %! $ % sin x − sin x $%
$! cos x = t.
+ R(sin x, cos x) & %! $ % cos x − cos x $%
$! sin x = t.
+ R(sin x, cos x) && $ ) % sin x
− sin x, cos x − cos x, $ && $! tg x = t.
$
dx
, I =
.
a2 cos2 x + b2 sin2 x
!
"
tg x = t =⇒ x = arctg t =⇒ dx =
#
t
1
sin x = √
, cos x = √
,
2
1+t
1 + t2
t2
1
, cos2 x =
.
sin2 x =
1 + t2
1 + t2
tg x = t,
$
$
I=
=
dt
.
1 + t2
dt
1+t2
a2
1+t2
+
b2 t2
1+t2
$
=
dt
1
=
a2 + b2 t2
b
bt
1
1
arctg + C =
arctg
ab
a
ab
%&!'
$
bdt
=
a2 + (bt)2
b
tg x + C.
a
$
I=
dx
.
sin 3 x cos2 x
$ sin x − sin x
cos x = t =⇒ sin x =
√
dt
1 − t2 =⇒ x = arccos t =⇒ dx = − √
.
1 − t2
(
$
I=−
dt
=−
√
2
1 − t (1 − t2 )3 t2
) *
$
dt
.
(1 − t2 )2 t2
+! +
1
(1 − t2 )2 t2
, +
I=
cos x
1
3 x
−
+ ln tg + C.
cos x 2 sin2 x 2
2
#
tg n xdx
1
−1
cos2 x
n
tg x = t
x = arctg t
# 4
I = tg 2xdx
tg 2 x =
!
"
dx =
#$
% & ' ( tg 2x = t" ) x = arctg t dx =
*+
1
2
dt
1+t2
1 dt
2 1+t2
$ 4
$
t dt
1
1
1
=
t2 − 1 + 2
dt =
I=
2
t2 + 1
2
t +1
t 1
tg3 2x tg 2x 1
t3
−
+ arctg tg 2x + C =
= − + arctg t + C =
6
2 2
6
2
2
tg3 2x tg 2x
−
+ x + C.
=
6
2
#
ctg n xdx
#
R(tg x) dx
tg x = t
t
$
tg x + 3
dx.
I = tg
x−1
, )( - )
. + ( !
/ )
##
% & . ! !
t = tg x =⇒ x = arctg t =⇒ dx =
( )
dt
,
1 + t2
( ) $ / /
I=
t+3
dt.
(t − 1)(t2 + 1)
% ) /&- /
t+3
A
Bt + C
=
+ 2
.
(t − 1)(t2 + 1)
t−1
t +1
t + 3 = A(t2 + 1) + (Bt + C)(t − 1).
t = 1 A = 2
! t2 ! ! B = −2
C = −1.
" #
2t + 1
2
−
dt =
I=
t − 1 t2 + 1
$
$
$
dt
d(t2 + 1)
dt
=2
−
−
=
t−1
t2 + 1
t2 + 1
= 2 ln |t − 1| − ln(t2 + 1) − arctg t + C.
$
" !
1
tg 2 t + 1 =
arctg tg t = t,
cos2 t
I = 2 ln | tg x − 1| + ln | cos x| − x + C.
$ ! % &
& % $ ! & ! & !
%
'( ) I = # cos 3x cos 9x dx.
* + #
$
$
1
1
(cos(−6x) + cos 12x) dx =
(cos 6x + cos 12x) dx =
I=
2
2
1
1
sin 6x +
sin 12x + C.
=
12
24
'( ,
I=
#
sin 2x cos 5x sin 9x dx.
$
$
1
1
(− sin 3x + sin 7x) sin 9x dx = −
sin 3x sin 9x dx+
I=
2
2
$
$
$
1
1
1
+
sin 7x sin 9x dx = −
(cos 6x − cos 12x) dx +
(cos 2x−
2
4
4
1 sin 12x sin 6x sin 2x sin 16x
− cos 16x) dx =
−
+
−
+ C.
4
12
6
2
16
#
sinn x cosm x dx
!
" #
$%& I = # sin2 x cos7 x dx.
$
$
I = sin2 x cos6 x cos x dx = sin2 x(1 − sin2 x)3 d sin x =
$
= (sin2 x − 3 sin4 x + 3 sin6 x − sin8 x) dx =
=
sin3 x 3 sin5 x 3 sin7 x sin9 x
−
+
−
+ C.
3
5
7
9
$%$
$
I=
sin5 x
√
dx.
cos x
' "
(
− sin x dx = dt
cos x = t!
$
$
−1/2
(1 − t2 )2
√
+ 2t3/2 − t7/2 dx =
dt =
−t
t
√
√
4 2√
2 √
4
2
= −2 t + t t − t4 t + C = cos x(−2 + cos2 x − cos4 x) + C.
5
9
5
9
I=−
) * +!
! "#
$%% I = # cos4 x dx.
2
1 + cos 2x
,
cos x = (cos x) =
2
$
$
1
1
(1 + cos 2x)2 dx =
(1 + 2 cos 2x + cos2 2x) dx.
I=
4
4
4
2
2
cos2 2x =
1+cos 4x
2
$
1 + cos 4x
1
1 + 2 cos 2x +
dx =
4
2
$
1
3
1
1 3
1
=
( + 2 cos 2x + cos 4x) dx = ( x + sin 2x + sin 8x) + C.
4
2
2
4 2
8
I=
I=
$
#
cos2 3x sin4 3x dx.
$
sin2 6x 1 − cos 6x
dx =
4
2
$
$
1
1 − cos 12x
1
=
(sin2 6x − sin2 6x cos 6x) dx =
−
8
8
2
1 x sin 12x sin3 6x
− sin2 6x cos 6x dx =
−
−
+ C.
8 2
24
18
I=
(cos 3x sin 3x)2 sin2 3x dx =
! " # $ % &
'm + n = 2k k ∈ N
( ) (# m n (# "# ( &
" # ( #
$
*
I=
3
cos2 x
dx.
sin8 x
+ ,
m = − 83
2
n = 3
m + n = − 83 + 23 = −2
$ ctg x = t )$
dx
− 2 = dt
sin x
3
2
cos2 x
= ctg 3 x.
2
sin x
$
I=−
2
3 5
3
t 3 dt = − t 3 + C = − ctg x 3 ctg 2 x + C.
5
5
n tg x ctg x
I = # tg4 x dx.
! " #
dt
=
I = tg x = t, x = arctg t, dx = 2
t + 1
$
$ 4
1
t dt
=
dt =
=
t2 − 1 + 2
t2 + 1
t +1
tg3 x
t3
− tg x+
= − t + arctg t + C =
3
3
3
tg x
+ arctg(tg x) + C =
− tg x + x + C.
3
$
I=
#
ctg 5 x dx.
! " # %& '
1
dx
ctg 2 x =
−1
= − d ctg x :
2
2
sin
x
sin
$x
$
1
−
1
dx =
I = ctg 3 x ctg 2 x dx = ctg 3 x
sin2 x
$
$
ctg 4 x
−
= − ctg 3 x d ctg x − ctg 3 x dx = −
4
$
$
1
ctg 4 x
− ctg x
+ ctg x d ctg x+
− 1 dx = −
2
4
sin x
$
ctg 4 x ctg 2 x
+
+ ln | sin x| + C.
+ ctg x dx = −
4
2
() ) " %
*
$
+, I = sindx3 x .
x
2t
dt
=
I = tg = t, sin x =
, dx =
2
1 + t2
1 + t2
$
$
1
(1 + t2 )2
1
1
2
=
dt =
+ + t dt =
4
t3
4
t3
t
1
t2
1
− 2 + 2 ln |t| +
+C =
=
4
2t
2
x 1
x
1
x
1
− ctg2 + ln | tg | + tg2 + C.
8
2 2
2
8
2
#
cosdx x
5 π
5
cos x sin 2 + x
5
! ""
$
I=
5 + 6 sin x
dx.
sin x(4 + 3 cos x)
#$ tg x2 = t
$
I=
2t
1+t2
5 + 12t2
2 dt
1+t 2
=
3(1−t ) 1 + t2
4 + 1+t2
$
5t2 + 12t + 5
dt.
t(7 + t2 )
%
5t2 + 12t + 5
A Bt + C
= +
.
t(7 + t2 )
t
7 + t2
#
5t2 + 12t + 5 = A(7 + t2 ) + t(Bt + C);
5
30
A = , B = , C = 12.
7
7
&
30
t + 12
5
dt =
+ 7
7t
7 + t2
5
15
12
t
= ln |t| +
ln(7 + t2 ) + √ arctg √ + C =
7
7
7
7
12
x
x
1
x
5
ln | tg | + 3 ln(7 + tg2 ) + √ arctg( √ tg ) + C.
=
7
2
2
7
7 2
$
I=
I=
I=
I=
I=
I=
I=
#
#
#
#
#
#
$
sin 6x cos 2x dx.
cos 2x cos 3x dx.
sin5 x cos2 x dx.
√
3
cos5 x sin2 x dx.
sin4 x dx.
sin4 x cos6 x dx.
dx
.
sin4 x cos6 x
$
sin x
dx.
I=
cos9 x
$
I = ctg5 x dx.
I=
$
I=
I=
$
tg8 x dx.
dx
.
sin5 x
! " #
$
√
#
R(x; n ax + b) dx
√
√
% R(x; n ax + b)dx
$ R(x; n ax + b) &
√
x n ax + b n
#
' $ "
(
√
ntn−1
tn − b
, dx =
dt, ax + b = t.
ax + b = tn , x =
)*+,a
a
. $
n
$
$
√
n
R(x; ax + b)dx = R
n−1
nt
tn − b
;t
dt.
a
a
% "
$
*+,
' ( / (
x + 1 = t2 , x = t2 − 1
/0 (
$
I=
t · 2tdt
=2
t2 − t − 1
# t+1
$
√
x+1
√
dx
x− x+1
$
I=
t2 dt
=2
2
t −t−1
dx = 2tdt.
$
1+
t+1
t2 − t + 1
dt.
% t −t−1 " 0
" #$
1 2
1
(t − t − 1) = z, t = z +
dt = dz.
2
2
1 ' ' (
2
$
$
$
z + 32
d(z 2 − 54 )
dz
dz
=
2t
+
+
3
=
5
5
2
2
2
z −4
z −4
z − 54
√
z − 5
3
5
2
2
√ + C = t + ln |t − t − 1|+
= t + ln z 2 − + √ ln
4
5 z + 25
√
t − 1 − 5
3
2
2
√ +C =
+ √ ln
5 t − 12 + 25
√
2 x + 1 − 1 − √5
√
√
3
√ + C.
= x + 1 + ln |x − x + 1| + √ ln √
5 2 x + 1 − 1 + 5
I = 2t + 2
$
R x;
n
ax + b
cx + d
dx
ax + b
= tn .
cx + d
I =
$
x
x−1
dx
x+2
! "
x−1
= t2 ,
x+2
" #
$
I=
1 + 2t2
x−1
= t, x =
x+2
1 − t2
6t
1 + 2t2
t
dt = −6
1 − t2 (1 − t2 )2
$
dx = (1 −6tt2)2 dt.
2t4 + t2 dt
.
(t2 − 1)3
$ $ ! %
2t4 + t2
2t4 + t2
=
=
2
3
(t − 1) )
(t − 1)3 )(t + 1)3
B
E
C
D
F
A
+
+
+
+
+
.
=
t − 1 (t − 1)2 (t − 1)3 t + 1 (t + 1)2 (t + 1)3
2t4 + t2
=
A(t − 1)2 (t + 1)3 + B(t − 1)(t + 1)3 + C(t + 1)3 +
(t − 1)3 )(t + 1)3
+ D(t − 1)3 (t + 1)2 + E(t − 1)3 (t + 1) + F (t − 1)3 / (t − 1)3 (t + 1)3 .
t = 1 : 3 = 8C ⇒ C = 3/8
t = −1 : 3 = −8F ⇒ F = −3/8
"##$
!
2t4 + t2 = A(t2 − 1)(t + 1) + B(t2 − 1)(t + 1)2 + C(t + 1)3 +
+ D(t2 − 1)2 (t − 1) + E(t2 − 1)(t − 1)2 + F (t − 1)3 =
= A(t4 − 2t2 + 1)(t + 1) + B(t2 − 1)(t2 + 2t + 1) + C(t3 + 3t2 + 3t + 1)+
+ D(t4 − 2t2 + 1)(t − 1) + E(t2 − 1)(t2 − 2t + 1) + F (t3 − 3t2 + 3t − 1).
%
2t4 + t2 = A(t5 + t4 − 2t3 − 2t2 + t + 1) + B(t4 + 2t3 − 2t − 1)+
+ C(t3 + 3t2 + 3t + 1) + D(t5 − t4 − 2t3 + 2t2 − t − 1)+
+ E(t4 − 2t3 + 2t − 1) + F (t3 − 3t2 + 3t − 1).
"##$ & & !
&
⎧
t5 :
⎪
⎪
⎪
⎨ t4 :
⎪ t3 :
⎪
⎪
⎩
t:
A + D = 0,
A + B − D + E = 2,
−2A + 2B + C − 2D − 2E + F = 0,
A − B + C − D − E − F = 0.
" C = 38 F
= − 38
⎧
A + D = 0,
⎪
⎪
⎪
⎨ A + B − D + E = 2,
⎪ −A + B − D − E = 0,
⎪
⎪
⎩
A − B − D − E = − 34 .
' A =
5
;
16
B =
11
;
16
5
D = − 16
; E =
11
16
$
t8 dt
1
= 12
t7 + t6 + t5 + t4 + t3 + t2 + t + 1 +
dt =
t−1
t−1
t8 t7 t6 t5 t4 t3 t2
= 12
+ + + + + + + t + ln |t − 1| + C.
8
7
6
5
4
3
2
√
! t = 12 x + 1
√
3
12
12
(x + 1)2
(x + 1)7
(x + 1)5
x+1
+
+
+
+
I = 12(
8√
7
6
5
√
√
3
4
6
√
√
x+1
x+1
x+1
+
+
+ 12 x + 1 + ln | 12 x + 1 − 1|) + C.
+
4
3
2
√
#
R(x; Ax2 + Bx + C) dx
" 12 (Ax2 + Bx + C) = t #
$
12
$ %
√
√
√
a2 − t2 , t2 − a2 , a2 + t2 .
&'(')
* # # $ %
&'(') !
√
a2 − t2 =⇒ t = a sin z t = a cos z,
√
a
a
,
t2 − a2 =⇒ t =
t=
sin z
cos z
√
a2 + t2 =⇒ t = a tg z t = a ctg z.
'('
I =
+ , "
1 2
(x + 2x − 3) = t x + 1 = t,
2
,
$ √ 2
x + 2x − 3
dx
(x + 1)3
x = t − 1, dx = dt
$
$ √2
(t − 1)2 + 2(t − 1) − 3
t −4
I=
dt =
dt.
t3
t3
- $
t=
2
,
cos z
√
t2 − 4 = 2 tg z, dt =
2 sin z
dz.
cos2 z
$
$ √2
$
1
t −4
2 tg z 2 sin z
dz
=
sin2 z dz =
dt
=
t3
2
( cos2 z )3 cos2 z
$
1
sin 2z
1
1
(1 − cos 2z)dz =
z−
+ C = (z − sin z cos z) + C.
=
4
4
2
4
I=
2
2
2
, cos z = , z = arccos ,
cos z
t√
t
2−4
√
t
.
sin z = 1 − cos2 z =
t
t=
√
2 2 t2 − 4
1
+C =
arccos −
4
t
t2
√
1
2
2 x2 + 2x − 3
=
arccos
−
+ C.
4
x+1
(x + 1)2
I=
!
" " # $%
%
I =
&'
$
I=
$
dx
√
( 5 + 2x + x2 )3
1 2
(x + 2x + 5) = t dx = dt
= 2
2
3
x+2=t
x= t−2
(5 + 2x + x )
dx
=
$
t = 2 tg z
dt
dt = cosdz2 z 1
√
=
=
=
2
4 cos zdz =
t2 + 4 = cos z
(t2 + 4)3
z
1
t
1
1
+C =
= sin z + C = sin arctg + C = 2
4
4
2
4 z2 + 1
$
4
x+1
+ C.
= √
4 5 + 2x + x2
$
dx
√
x>m
(x − m) ax2 + bx + c
dt
1
x − m = , dx = − 2 .
t
t
$
I = √ 2 dx
x 5x − 2x + 1
! " # $ %#
$
d(t − 1)
dt
√
=−
=
2
1
5
2
t − 2t + 5
(t − 1)2 + 4
−
+
1
2
t
t
t
1
√
1
2
2
+
5
+
= − ln |t − 1 + t + 2t + 5| + C = − ln − 1 +
+ C.
x
x2 x
$
I=
− dt
t2
$
=−
& ' #
$
xm (a + bxn )p dx,
m, n p ( ' %
) % *+ % + ,
% , -
. p ( % / , % +% a + bxn 0
p xm %
( % / , %
1 m+1
n
a + bxn = ts , s ( p
+ p ( % / , %
m+1
n
−n
ax + b = ts , s ( p
2
$
3
√
1+ 4x
√
dx.
x
$
1/3
x−1/2 1 + x1/4
dx.
I=
1
1
m+1
1
=⇒
= 2 −
m=− , n= , p=
2
4
3
n
1+x
1/4
= t3 =⇒ t =
3
.
1 + x1/4 =⇒
−1/2
=
x1/4 = t3 − 1 =⇒ x = (t3 − 1)4 =⇒ x−1/2 = (t3 − 1)4
1
=⇒ dx = 4(t3 − 1)3 3t2 dt = 12(t3 − 1)3 t2 dt
= 3
(t − 1)2
$
$
t · 12(t3 − 1)3 t2 dt
=
12
(t3 − 1)2
$
t3
−
= 12 (t6 − t3 )dt = 12t4
7
I=
t3 (t3 − 1)dt =
1
+ C.
4
!
t=
I = 12(1 +
√
4
3
x)
3
1+
1+
√
4
x
√
4
x,
√
1+ 4x 1
+ C.
−
7
4
" #$ %& %& '
& %& %& ( !
' ) %& **
( + ,#$*-. ,#$*#.*
' u = u(x; y) !
( ' u = Ref (z)
f (z) = u(x; y) + iv(x; y)* /% ( '
v = v(x; y) = Imf (z) '( f (z)*
∂u
=
⇒v =
1 ∂v
∂y
∂x
ϕ(x) x
2 !" """
# ∂u
∂v
∂u
=
dx + ϕ (x) = − " #"
∂x
∂x
∂y
∂u
dx + C
∂y
x
$" % & ' ϕ(x)
# ∂u
dx + ϕ(x)
v = Imf (z) =
∂x
ϕ(x) = −
#
x
# ∂u
dx
∂x
# ∂u
dy+ϕ(x)
∂x
ϕ (x) "
+
v(x) ( %"
)% '" *&" " % *
"% ( ! ( %"
+, u(x; y) = x3 − 3xy2
f (z)
f (z) f (0) = 0
- . / *& %" u(x; y) =
" % ( 0" '" *&" ("
" ( %" 1 "% ( u(x; y) = x3 − 3xy2 =
= x3 −3xy 2
#
∂v
∂u
=
= 3x2 − 3y 2 ⇒ v(x; y) = 3 (x2 − y 2 )dy + ϕ(x) =
∂y
∂x
∂v
∂u
2
3
= 6xy + ϕ (x) = −
= 6xy ⇒ ϕ (x) = 0 ⇒
= 3x y − y + ϕ(x)
∂x
∂y
⇒ ϕ(x) = C
" v(x; y) = Imf (z) = 3x2y − y3 + C f (z) = x3 − 3xy2 + i(3x2y −
− y 3 + C) "" C # % f (0) = 0 "
z = x + iy = 0 ⇒ x = y = 0 f (0) = 0 + iC = 0 ⇒ C = 0
2 %" f (z) = x3 − 3xy2 + i(3x2y − y3 )
= Ref (z) ⇒
" 3 " ' *4 %
% "% " ! " %!"
"
. " " " % 1%" '
" " & '" * % 0"
& "
$
$
$
2
ex dx,
√
sin x
dx,
x
x cos x dx
! "
#$ %
$
I=
sin x cos x dx.
& ' ( )
sin x cos x *
$
1
sin x d(sin x) = sin2 x + C,
2
$
1
I = − cos x d(cos x) = − cos2 x + C,
2
$
$
1
1
1
sin 2x dx =
sin 2x d(2x) = − cos 2x + C.
I=
2
4
4
I=
#$ +
!
1
1 2
sin x, − cos2 x
− 41 cos 2x
2
2
" # #
, ' . "
'
) " .
$
−6x2 − 5x + 8
dx
x3 + 2x2 − 1
## # ' /0
1
$
3x2 + 4x
dx = ln |x3 + 2x2 − 1| + C
x3 + 2x2 − 1
! "# "$
% && ' (
! ) * + ) , - %
. &/0&$
. 1
1 1
"2 &
+%
$
3
I=
3
I = |x − 2 = t6 , dx = 6t5 dt| =
√
3
x−2
dx.
√
(x − 2)2 − x − 2
$
t4
t2
6t5 dt =
− t3
1
dt =
=6
t +t +t+1+
t−1
t4 t3 t2
=6
+ + + t + ln |t − 1| + C =
4
3
2
√
√
3
3
(x − 2)2
x−2
x−2
+
+
+
=6
4
3
2
$
t4
dt = 6
t−1
$
3
2
√
√
+ 6 x − 2 + ln | 6 x − 2 − 1| + C.
"2 &
√
$
$
3
(x − 2)2 − x − 2
√
I=
dx
=
(x − 2)1/3 −
3
x−2
3
6
−(x − 2)1/6 d(x − 2) = (x − 2)4/3 − (x − 2)7/6 + C.
4
7
ax+b
cx+d
$
I=
5 − 3x
dx.
4 + 7x
! " # $ 5−3x
= t2
4+7x
&
%
x dx#
5 − 4t2
⇒
7t2 + 3
2
2
−8t(7t + 3) − 14t(5 − 4t )
−94t
⇒ dx =
dt =
dt.
2
2
(7t + 3)
(7t2 + 3)2
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$'
$
$
−94t
t2 dt
dt
=
−94
=
2
2
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$
$
t +7−7
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1
dt
1
3
dt
=
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·
−
.
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49
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t2 + 37
(t2 + 37 )2
I=
t
$ ( ) * #
$
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dt
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2
3
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1
2
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t
3 +
2
2
t +7
2(t + 37 )
.
+ ' ( , ,
( (
√
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7 5 − 3x
−
arctg
+ C.
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7
147
3 4 + 7x
-
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3
dx
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.
3
(x − 1)2 (x + 2) =
(x−12 )
(x
(x+2)2
2
+ 2)3 = (x + 2) 3 x−1
.
x+2
x − 1
dx
=
= t3 ,
I=
x+2
3
x−1 2
(x + 2)
x+2
$
dt
2t3 + 1
9t2 dt
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x=
,
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3
2
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(1 − t )
1 − t3
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1
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1 − t t2 + t + 1
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1
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3
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1
2t + 1
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3
1
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2
√
√
√
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3
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23x−1+ 3x+2
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33x+2
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$
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1
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2
2
2
2
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1
dt
1
47
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2
2
t2 + 47
4
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1
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2
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3x − 7
dx.
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1
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t−4
2
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=
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2
5
5
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3
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5
dt =
2
8
5 5 t−4
+
(t
−
4)
+
1
5
5
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2
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2
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t − 11
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t − 11
√
3 √
47
= √ t2 − 11 − √ ln |t + t2 − 11| + C =
5 5
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√
3
47
=
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5
5 5
$
dx
√
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x + d = 1t
$
dx
√
.
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x x2 + 3
=
$
1
− dt/t2
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√
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=
t
t
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√
$
$
dt
1
d( 3t)
√
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= −√
=
3t2 + 1
3
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√
3 √x2 + 3
√
√
1
1
2
+
= − √ ln | 3t + 3t + 1| + C = − √ ln
+ C.
x
3
3 x
$
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dx
√
.
x2 7 − x2
$
$
dt
− dt/t2
1
t dt
√
I = x = , dx = − 2 =
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=
t
t
7t2 − 1/t3
7t2 − 1
$
1 √
1
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1 7
√
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=−
− 1 + C.
14
14
7 x2
7t2 − 1
$
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#$
$
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I=
dx
√
.
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1
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=
3
27/ cos t
9
$
cos t dt =
9
.
cos2 t
1
sin t + C.
9
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x
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3
tg t
x/3
x
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=
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.
2
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1 + (x/3)
1 + tg t
x
I = √
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x = 3 tg t ⇒ tg t =
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x dx
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3
x( x + 4 x)
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(x + 1)2 (x − 2)4
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$
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I=
I=
I=
I=
√
dx
.
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√
2x + 5
dx.
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$
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dx
√
.
x 2x2 − 1
dx
√
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dx
√
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$
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x(3 + 4x3 ) dx.
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$
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%&'().*(
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2
x + 4x + 20
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8
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=
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=
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t+
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y
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0
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1
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t2 + 2t
2
t+2
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1
t dt
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t+2
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1
t+2−2
dt =
t+2
dt = 2 (t − 2 ln |t + 2|) |31 = 4 1 − ln
I=
$10
3
dx
√
.
(x − 4) x + 6
x + 6 = t2
t
x = 3 t = 3,
x = 10 t = 4.
$4
I=
3
5
.
3
dx = 2t dt
t − √10 4
2t dt
1
√ =
= 2 √ ln
(t2 − 10) t
2 10 t + 10
3
4 − √10
3 − √10
1
√ − ln
√ =
=√
ln
10
4 + 10
3 + 10
√
4 − √10 3 + √10
1
7 + 2 10
1
√
√ = √ ln
= √ ln
.
3
10 4 + 10 3 − 10
10
I=
#2a √
2ax − x2 dx.
0
12 (2ax − x2) = t
a−x = t x = a−t dx = − dt 2ax−x2 = a2 −t2 x = 0 ⇒ t = a
x = 2a ⇒ t = −a
$a √
I =
a2 − t2 dt.
−a
t = a sin z dt = a cos z dz
= a cos z t = −a ⇒ z = − π2 t = a ⇒ z =
2
$π/2
I=a
π
2
√
a2 − t2 =
$π/2
cos z dz = a
(1 + cos 2z) dz =
2
2
0
−π/2
2
=a
π/2
πa2
sin 2z
.
=
z+
2
2
0
z
[− π2 ; π2 ]
!
" " "
#
$π/2
I=
$π/2
cos z dz = 2 cos2 z dz.
2
0
−π/2
!
$ "
# $
u = x;
du = dx
I =
dv = ex dx; v = ex
I=
#1
xex dx.
0
"
$1
= xex |1 − ex dx = e − ex |1 = 1.
0
0
0
"
#
%
$
$ "
I=
#1
x arctg x dx.
0
u = arctg x; du = 2dx
x +1
I =
2
dv = x dx; v = x2
=
=
1
$1 2
$1
x dx
π 1
1
x2
arctg x −
=
−
2
2
x2 + 1
8 2
0
0
1−
0
1
x2 + 1
dx =
π 1
π π 1
π 1
= − (x − arctg x)|10 = −
1−
= − .
8 2
8 2
4
4 2
f (x) [a; b]
n
ξi
y = ex [a; b] n
ξi
# x dx !
b
a
f (x) = x " [a; b] n
# ξi
I=
$1/2
√
0
! I =
#3
2
√
0
ex dx
$1
" I =
dx
1 − x2
dx
x2 + 4
$4 √
I=
1
x dx
√ .
1+ x
y
b
a x
0
1
b
S=
4
a
$a √
a2 − x2 dx.
0
1
S = |x = a sin t, x = 0 =⇒ t = 0, x = a =⇒ t = π/2| =
4
$π/2
$π/2
1 + cos 2t
b 2
1
1
πab
π/2
2
= ·a
dt = ab( t + sin 2t)|0 =
.
cos t dt = ab
a
2
2
4
4
0
0
! ! " #
S = πab.
! a = b = R
S = πR2 .
$ %! !
! % ! & $" ! !'
! ( ! ) " & * * +! ,
- & " $
a x b,
y (x) y (x).
. ! ! ( !
$b
(y (x) − y (x))dx.
SABCD =
a
+/0,
y
y=yb (x)
D
C
B
A
y=yH (x)
0
a
x
b
y (x) y (x) x ∈ [a; b].
! " #
x = x(t),
y = y(t),
$t2
S=
y(t)x (t) dt.
$%&'
t1
" ! $(&)*
y = f (x) = y(t), dx = x(t) dt& +" t1
, a, t2 - , b&
$%&'
x = a(t − sin t),
y = a(1 − cos t)
. # /, ,
!" t1 = 0 t2 = 2π & $0&
y
a
a
t=2π
t=0
x
$2π
$2π
a(1 − cos t)(a(t − sin t)) dt = a
(1 − cos t)2 dt =
2
0
0
$2π
$2π
1 + cos 2t
= a2 (1 − 2 cos t + cos2 t) dt = a2
1 − 2 cos t +
dt =
2
0
0
2π
3
sin 2t
2
t − 2 sin t +
= a2
= 3πa .
2
4
0
r = r(ϕ) ! "
OAB # $$%
A
r
+d
r
α
O
r=r(ϕ)
dϕ
r
B
β
r
dϕ
! " dS = 12 r2 dϕ,
1
S=
2
$β
#$%!&'
r2 dϕ.
α
$%!& r = cos 3ϕ
π/6
r
( )
S =2·
=
1
2
1
2
$π/6
cos2 3ϕ dϕ =
0
$π/6
(1 + cos 6ϕ) dϕ =
0
1
2
ϕ+
π/6
sin 6ϕ
π
= 12 .
6
0
" *+ V ! "
,
Ox.
x : S = S(x).
a b
[a, b] n
a = x0 < x1 < x2 < · · · < xn−1 < xn = b.
!
Ox.
"
n #$
% &
$
xi−1 xi Δvi . ' %
V = Δv1 + Δv2 + · · · + Δvn =
n
Δvi .
i=1
&
$ xi−1 xi . (% #$ &$ #
$
% % [x1 − xi−1
ξi ∈ [xi−1 , xi]
#$ %
)
* s(ξi )Δxi.
! #$ &$ *
V ≈
n
+,-,.
s(ξi )Δxi .
i=1
/ #$
%
+,-,. *
V = lim
λ→0
n
+,-0.
s(ξi )Δxi.
i=1
*
$b
V =
+,-1.
s(x) dx.
a
,-,
2
2
2
xa2 + yb2 + zc2
= 1.
2 *
)
Ox ($ x = d (d ∈ [−a; a]) !
x = d
√
z2
y2
+
− d2 )
b2
(a2
a2
a1 = ab
c2
(a2 − d2 )
a2
√
c
b1 = a a2 − d2
a2 − d2
=1
πbc 2
(a − d2 ).
a2
d = x (−a x a)
S = πa1 b1 =
!"
S(x) =
# $%&
!
πbc 2
(a − x2 ).
a2
'"
$a
V =
$a
S(x) dx = 2π
−a
0
bc
= 2π 2
a
bc 2
(a − x2 ) dx =
a2
a
x3
4
a x−
= πabc.
3 0 3
2
(
a = b = c = R
R
a b c
V = 43 πR3
) * ! ' * * Ox
+ * , - $./ *0
/ $! , -1 S .* x
! * / ! *!
S(x) = πy 2 .
)** . * ',
$b
Vx = π
a
y 2 dx.
2
y
y=f(x)
O
a
dS
b
x
y = (x − 1)2, x = 0, y = 0
x y y x
1
$1
π
(x − 1)5
4
Vx = π (x − 1) dx = π
= 5
5
0
0
$1
Vy = π
√
( y + 1)2 dy = π
0
1
y 2 4 3/2
17π
+ y + y =
2
3
6
0
! L " " A " B
# $ %& ' " A1 , A2 , . . . , An−1
(
! A = A0 ) B = An ! " )
* " + A0 A1 A2 . . . An−1 An
, L !
"
! # !
$ ! ! # %
-* ! ' AB
+ A0 A1 . . . An
y
B
A i-1
A1
A n-1
Ai
A
x
n
Δli ,
i=1
Δli Ai−1Ai
Δli =
(xi − xi−1 )2 + (yi − yi−1 )2 = Δx2i + Δyi2 .
[xi−1; xi]
f (xi ) − f (xi−1 ) = f (ξi )(xi − xi−1 )
yi − yi−1 = f (ξi )(xi − xi−1 )
Δx2i + Δyi2 =
Δyi = f (ξi )Δxi
Δx2i + f 2 (xi )Δx2i =
1 + f 2 (ξi )Δxi .
ξi [xi−1; xi]
!
"# $ "" ξi
% "#& " ξi '
"# (
n
Δli =
i=1
n
1 + f 2 (ξi )Δxi .
[a; b]
l = lim
1+
n→+∞
i=1
f 2 (ξi )Δxi
$b
=
1 + f 2 (x)
1 + y 2 dx.
a
!"#
1 + y 2 dx.
$%
"& & &"'
'
*
( & ) & x=x(t),
y=y(t)
dl =
dl =
dx2 + dy 2 =
x 2 dt2 + y 2 dt2 = x 2 + y 2 dt
& ) & # +"
$t2
l=
x 2 (t) + y 2 (t) dt.
t1
( # & $%
"
x = r cos ϕ,
y = r sin ϕ,
dx = (r cos ϕ)ϕ dϕ = (r cos ϕ − r sin ϕ) dϕ,
dy = (r sin ϕ)ϕ dϕ = (r sin ϕ + r cos ϕ) dϕ.
& & " " & $%* )
dl = r2 + r 2 dϕ,
& "' ' &
l=
$β
α
r2 + r 2 dϕ.
$$
√ y = ln x
3.
√
$ 3√
1 + x2
dx.
L=
x
1
x = tg t =⇒ dx =
$π/3
L=
π/4
=2−
dt
=
sin t cos2 t
√
$π/3
dt
,
cos2 t
sin2 t + cos2 t
dt =
sin t cos2 t
t π/3
1
+ ln tg =
cos t
2
π/4
π/4
√
π
= 0,92.
2 − ln
3 tg
8
$2π
l=a
(t − sin t) 2 + (1 − cos t) 2 dt =
0
$2π
(1 − cos t)2 + sin2 t dt =
=a
0
$2π
$2π
$2π
t
2 t
2(1 − cos t) dt = a
4 sin dt = 2a sin dt =
=a
2
2
0
0
0
2π
t
= −4a cos = −4a(−1 − 1) = 8a.
2 0
L
!" u = f (x; y) # $%& "
L
y = y(x) #'
n
f (ξi ; y(ξi ))Δli =
i=1
n
f (ξi ; y(ξi )) 1 + y 2 (ξi )Δxi.
i=1
Δli → 0 n → +∞
u = f (x; y)
!
$
$
f (x; y(x))dl =
L
f (x; y(x))dl =
(maxΔli →0)
AB
""# dl =
L
f (ξi ; y(ξi ))Δli,
i=1
1 + y 2 (x) dx
x = x(t) y = y(t)
x = r cos ϕ y = r sin ϕ
!
"!
$ %%& !
$
$tB
f (x; y) dl =
AB
tA
$
$ϕB
f (x; y) dl =
#
' %! (
f (x(t), y(t)) x 2 (t) + y 2 (t) dt
f (r cos ϕ; r sin ϕ)
r2 (ϕ) + r 2 (ϕ) dϕ,
'
ϕA
AB
n
lim
n→+∞
tA tB ϕA ϕB
A B
t
!
ϕ
$ %
x = R cos t y = R sin t A(0; R) B(R; 0)
" # f (x; y) = x · y
) *
( + %!
$
$tB
f (x; y) dl =
AB
xy
tA
x 2 (t) + y 2 (t) dt.
0 = R cos t
tB = π2 R = R cos t
tA = 0 xt = −R sin t yt = R cos t
π
$
$2
f (x; y) dl =
R cos t · R sin t R2 sin2 t + R2 cos2 t dt =
0
AB
π
= R3
$2
sin t cos t dt = R3
0
π
sin2 t 2 R2
.
=
2 0
2
A B
tA < tB
AB
!
" γ = f (x; y) # " f (x; y)
P (x; y)
$ AB
%& P " '
() * %+ !
Δmi Ai−1Ai ' ,-) Δmi ≈ f (xi; yi)Δli
%+ AB m ≈
n
≈
f (xi; yi )Δli *
!
i=1
,./ !
$
m=
f (x; y) dl.
',.01)
AB
2 f (x; y) = 1 ,.01
,.3
AB
,.3
√
y = x2 x ∈ [0; 2]
γ = 3x
4 $ ,.01 y = 2x
√
√
0
0
$2 √
$2
3 √
2
m = 3x 1 + 4x dx =
1 + 4x2 d(1 + 4x2 ) =
8
√
3 2 1
3 2
1 + 4x2 2 = (93/2 − 1) = 6,5.
= ·
8 3
4
0
y
B
dl
y=f(x)
A
O
a
dx
b
x
Ox
! !" # $
% & '()*
dσ = 2πy dl,
dx
$b
σ = 2π
y dl = 2π
a
$b
y 1 + y 2 dx.
&'( +,)
a
'( +-
. * / ! $ ! %
x2 + y2 = R2 Ox. 0 1"
% '( ,
y 2 = R 2 − x2 .
$R
V =π
−R
2
$R
y dx = π
−R
−R
R
4
(R2 − x2 )dx = πR3 .
3
!
x
y = − , 1 + (y )2 =
y
" #
x2
1+ 2 =
y
x2 + y 2
R
= .
y
y
$ %
$R
$R
R
2
σ = 2π y 1 + y ( )dx = 2π y dx =
y
−R
−R
2
= 2πRx|R
−R = 4πR .
& !
' ( (
) * +
I
t I = I(t) Q
T
& ) , & [0; T ] t0 = 0,
t1 , t2 , . . . , tn = T n # +
[t0 ; t1 ], [t1 ; t2 ], . . . , [tn−1 ; tn ].
-. Δti = ti − ti−1
/ 0 ! .
% % τi
1 2 $ Δti
* 3 I(t) . % % τi $
I(τi)
4 ( # $
)
*
$ ! #
ΔQi ≈ I(τi )Δτi,
Q≈
n
i=1
I(τi )Δτi .
Q
n
Q = n→+∞
lim
I(τi )Δτi .
λ→0
i=1
! λ " #$
% ! & '()
$T
Q=
I(t) dt.
0
*(+ L =
!" #$ %
, $ -
.
x / $ .
L − x (L − x)50
0 #
1
dx #
dA = −(L − x)50 dx.
-2 # !
$L
(L − x)2 L
A = −(L − x)50 dx = 50
=
2
0
0
= −25L2 = −25 · 1002 = −250000.
*+ & " %
'
*(3 (" %"
) V = t2 *" + %" $
,
,
$
$2
S=
0
2
t3
t dt = = 1.
8 0
2
√
y=2 x
y = 2x
y
y=2x
C 2
A
y=2 x
m
O
B
1
x
! "
Ox − OmAB OAB
$b
$1
(y − y )dx =
S=
√
2 x dx −
0
a
$1
0
#
1
1 1
4 3/2
2x dx = x − x2 = .
3
3
0
0
! "
#
Oy − OAC
OmAC $ % &
!
' ( " ) "
( "
$2
$2
(x − x )dy =
S=
0
0
y
dy −
2
$2
0
2
2
y2
y3
1
y2
dy = − = .
4
4 0 12 0 3
x = a cos t,
y = b sin t.
! " # "
$t2
S=
y(t)x (t) dt.
t1
y
t=π/2 b
-a
O
a
t=0
x
-b
$ % " " #&% ## ' (
) " " " y = b sin t, dx = d(a cos t) = −a sin t dt t ∈ [ π2 ; 0] #
$0
S = −4ab
sin2 t dt = −2ab
π/2
= −2ab(t −
$0
(1 − cos 2t) dt =
π/2
0
1
sin 2t) = πab.
2
π/2
$ # * # ! +
' " # , " # ,
-
! " r = a sin 2ϕ #
α = 0 β = π2
S=
1
2
$β
ρ2 dϕ =
a2
2
α
r = a sin 2ϕ
$π/2
sin2 2ϕ dϕ =
0
π/2
$π/2
1
a2
πa2
a2
.
(1 − cos 4ϕ) dϕ = (ϕ − sin 4ϕ) =
=
4
4
4
8
0
0
r
y = 2√x y = 2x Ox Oy
! " # $%
V2 & ' V1 ( ) * OmAB + V2 (
' OAB Ox
, ' +
V1
$1
Vx = V1 − V2 = π
4x dx − π
0
$1
2
4x dx = π
0
1
2x2 0
) Oy
# /% ! x y y x
$b
Vy = π
a
x2 dy.
1
2
x3
− 4 = π.
3 0
3
- .
$2
Vy = π
0
y2
dy − π
4
$2
0
y4
dy = π
16
2
2
4π
y 5
y 3
=
.
−
12 0 80 0
15
R
x2 +y2 = R2 Ox !" # $
√
x
R 2 − x2 , y = − ,
y
2
x2 + y 2
R
x
= .
1 + y2 = 1 + 2 =
y
y
y
y=
% $$ # $
$b
S = 2π y 1 + y 2 dx,
a
"
$R
S = 2π
y
−R
R
2
dx = 2πRx|R
−R = 4πR .
y
√
& y = ln x (x ∈ [1; 3])
' () ) )$
* + ,-
√
$ 3√
l=
1
$π/3
=
x = tg t
1 + x2
x =√1 ⇒ t =
dx =
dx = cosdt2 t x = 3 ⇒ t =
x
π/4
$π/3
=
π/4
$π/3
dt
=
sin t cos2 t
sin2 t + cos2 t
dt =
sin t cos2 t
π/4
sin t dt
+
cos2 t
$π/3
π/4
dt
=
sin2 t
t π/3
1
+ ln tg =
cos t
2
π/4
π
4
π
3
=
√
√
π
1
1
= 2 − 2 − ln 3 − ln tg = 2 − 2 −
2
8
2
√
√
1 3( 2 − 1)
≈ 0,91.
= 2 − 2 − ln √
2
2+1
1 − cos π4
ln 3 + ln
1 + cos π4
=
R
x = R cos t,
y = R sin t,
l=
#t2 2
x + y 2 dt =
t1
#2π
#2π
=
(R cos t) 2 + (R sin t) 2 dt = R dt = 2πR.
0
0
r = 1 − sin ϕ.
r = − cos ϕ!
r2 + r 2 = (1 − sin ϕ)2 + cos2 ϕ = 2(1 − sin ϕ) =
π ϕ
π ϕ
π
+ ϕ = 4 cos2
+
= 2 cos
+
.
= 2 1 + cos
2
4
2
4
2
" #!
$π/2
l=2
−π/2
2 cos
π
4
+
π ϕ π/2
ϕ
dϕ = 8 sin
+
= 8.
2
4
2 −π/2
$ % &' ( % % ' % '
* +
)
,
O
F = −kx k
! x " #
" $ "
x = a x = b
dx
dA
dA = −kx dx.
$b
F (x) dx = −
a
kx dx = −k
a
$b
A=
a b
k
x2
= (a2 − b2 ).
2
2
! y = 0
y = 4 − x
y = (x + 2)2
x = R cos3 4t ,
y = R sin3 4t .
"
r = a(1 + cos ϕ)
#
x = 0 y = 0 z = 0 x + y + z = 1
! x = const
!
y2 = x Ox x = C
$ "
y = x (y ∈ [0; 1])
Oy
%
y = 2x (x ∈ [0; 1]) Ox
& y = ax2 (a > 0)
# $ x
x = R cos3 4t ,
y = R sin3 4t .
I =
$+∞
1
dx
xα
$+∞
$b
dx
I=
= lim
x−α dx.
xα b→+∞
1
1
α = 1, ! ln x b → +∞ "
# !
$ α = −1
$b
+∞
1
I = lim
x−α dx = lim
% &'
.
b→+∞
b→+∞ (1 − α)xα−1 1
1
$ α < 1 # !
1
# !
$ α > 1 α−1
(& ) * +
y
α=1
y=1/xα
α1
,
" * !
.
/ 0 / )0*) % *)0 !
x > 1 . . * '
&
I =
$+∞
1
dx
1 + x2
!
$+∞
I=
1
dx
= lim
1 + x2 b→+∞
$b
1
b
dx
= lim arctg x =
2
b→+∞
1+x
1
= lim (arctg b − arctg 1) =
b→+∞
4
$+∞
I=
4
I=
π π
− .
2
4
$+∞
dx
√ = lim
x b→+∞
$b
4
π
.
4
dx
√ .
x
√
√
dx
√ = 2 lim x|b4 = 2 lim ( b − 2) = +∞.
b→+∞
b→+∞
x
!
" # $ #% & $ '
b
$
$
b
f (x) dx = lim
−∞
(
*
f (x) dx.
a→−∞
a
$
#% $ ) '
$+∞
$c
$+∞
f (x) dx =
f (x) dx +
f (x) dx,
−∞
−∞
c
+ , ) #
( * # '
,
$ * #
$+∞
I=
−∞
$0
I=
−∞
arctg x dx
+
1 + x2
$+∞
0
arctg x dx
.
1 + x2
arctg x dx
.
1 + x2
$0
−∞
arctg x dx
= lim
a→−∞
1 + x2
$0
0
arctg x dx
1
lim arctg 2 x =
=
2
1+x
2 a→−∞
a
a
=−
1
π2
lim arctg 2 a = − .
2 a→−∞
8
$+∞
0
arctg x dx
= lim
b→+∞
1 + x2
=
$b
b
2
arctg x dx
1
lim arctg x =
=
2
1+x
2 b→+∞
0
0
2
1
π2
lim arctg b = .
2 b→+∞
8
I =−
π2 π2
+
= 0.
8
8
!" #
!" $
$+∞
−∞
t = arctg x =⇒ dt =
' '(
dx
, x = −∞ =⇒
1 + x2
⇒ t = −π/2, x = +∞ =⇒ t = π/2.
$+∞
−∞
dx
=
1 + x2
dx
.
1 + x2
% &
'
)
$π/2
−π/2
π/2
dt = t
= π.
−π/2
$+∞
+∞
f (x) dx = F (x)
= F (+∞) − F (a),
a
a
F (+∞) = lim F (x).
x→+∞
f (x) [a, b)
b
$b
f (x) dx
a
!
" # $ % b % b − ε (ε > 0)
f (x) [a; b − ε]
$b−ε
& ' % (
f (x) dx
) *% +( '
$b
a
( ( ε → 0,
$b−ε
f (x) dx = lim
f (x) dx
ε→0
a
- ./
a
0 ! ( %
1 2 (2 ( $
- / & ( $ - 3
/
4
I =
$1
0
dx
.
x−1
1
5 6 , (
x−1
x = 1 7 % ( ! ( +
$1
$1−ε
1−ε
dx
dx
= lim
= lim ln |x − 1|
I=
= lim ln ε = −∞.
ε→0
x − 1 ε→0
x − 1 ε→0
0
0
0
f (x) (a; b]
#b
f (x) dx
a
$b
$b
f (x) dx = lim
ε→0
a+ε
a
!
f (x) dx
" # x = c # [a; b]
$b
$c
f (x) dx =
a
$b
f (x) dx +
a
$
f (x) dx.
c
% & " # '
( ) & "
$
$1
−1
dx
.
x2
* + , # " x = 0
#0
#1 dx
. ) /
,-' ' ' xdx
x
−1
0
" ) ( 0
2
$1
0
dx
= lim
ε→0
x2
$1
2
dx
1
1
= − lim |1ε = − lim 1 −
ε→0 x
ε→0
x2
ε
= +∞.
ε
(
$1
−1
dx
x2
1
$1
−1
1
dx
1
=
−
= −2.
x2
x −1
! " #
$
$ [a; +∞)
!
a
ϕ(x)
+∞
#
!"
#
f (x)
0 ϕ(x) f (x).
+∞
#
f (x) dx
"
ϕ(x) dx
"
a
ϕ(x) dx
+∞
#
+∞
#
a
f (x) dx;
a
%
# $ I =
$+∞
1
+∞
#
1
&
dx
x2 +1
dx
.
x3 + 1
' &
( )*
+∞
# dx
+ [1; +∞) x31+1 x21+1
x3 +1
,
1
) f (x) ϕ(x) [a; b)
0 ϕ(x) f (x),
% x = b &
#b
a
a
f (x) dx
a
#b
ϕ(x) dx
#b
#b
0
ϕ(x) dx
f (x) dx
$1
√
3
I=
0
dx
.
1 − x5
[0; 1)
x = 1 ! "
1
√
# [0; 1) $%
3
1−x
x = 1
& ' √
0 1 √ x5 x 1 − x5 1 − x
1
1
( 3 1 − x5 3 1 − x √
√
3
3
1−x
1−x5
1
) *
√
#
3
1−x5
1
[0; 1) √
3
1−x
$1
dx
√
+ I =
3
1−x
0
$1−ε
√
3
I = lim
ε→0
0
3
dx
3
= − lim ε2/3 − 1 = .
2 ε→0
2
1−x
) ' * '
'
+ ,, ' # ' . * & * / 0 1
% . /
u1 + u2 +, · · · , +un + · · · ,
x = 1, 2, · · · n, · · ·
f (x)
[1; +∞),
u1 = f (1), u2 = f (2), · · · , un = f (n), · · · .
$+∞
I=
f (x)dx.
1
!"
#
$%& ' ()*%*+ ' ()*%,+%
-%& ' ()*%*+ ' ()*%,+%
. ' /
1
1
1
1
+ α + α + ··· α + ··· ,
α
1
2
3
n
$+∞
dx
I =
.
xα
1
α > 1 α 1.
α > 1
α 1.
! " # $! ! %
$
& '( # ' !
!
0
I=
$+∞
e−αx dx.
0
$+∞
$b
−αx
e−αx dx =
I=
e
dx = lim
b→+∞
0
0
1
= − lim e−bx − 1 =
α b→+∞
1
bα
b→+∞ e
lim e−αb = lim
α > 0
b→+∞
b → +∞.
α > 0,
p < 0.
1
,
α
+∞,
= 0,
ebα −→ +∞
α = 0
α > 0
α1 . α 0
I=
$+∞
cos x dx.
0
+∞
I = sin x 0 = sin(+∞) − sin 0.
sin x x → +∞
! " # $
%
$1
dx
.
1 + x2
I=
−∞
I = arctg x 1−∞ = arctg 1 − arctg(−∞) = π/4 + π/2 = 3π/4.
&' " "
lim arctg x = π/2.
x→+∞
! " # $
$+∞
I=
−∞
3π/4.
arctg2 x dx
.
1 + x2
$0
I = I1 + I2 =
$0
I1 =
arctg 2 x d(arctg x) =
−∞
I2 =
−∞
arctg2 x dx
+
1 + x2
$+∞
0
arctg2 x dx
.
1 + x2
1
1
arctg 3 x 0−∞ = (0 − (−π/2)3 ) = π 3 /24,
3
3
$+∞
1
1
arctg 2 x d(arctg x) = arctg 3 x −∞
= ((π/2)3 − 0) = π 3 /24.
0
3
3
0
I = I1 + I2 = π 3 /24 + π 3 /24 = π 3 /12.
!
" #
$%&
$+∞
I=
0
dx
.
1 + x3
! '" [1; +∞) (")
" *
$%+
1
1
.
3
1+x
1 + x2
, - " . "
"/
$%0
I=
$+∞
2
e−x dx.
0
#
(x − 1)2 0 =⇒ x2 − 2x + 1 0 =⇒ −x2 −2x + 1.
! ex "
e−x e−2x+1 e−x e · e−2x .
#$% " &
2
2
I=
$+∞
e−2x dx
0
'
( '" )
!
#$*
$+∞
I=
1
arctg x dx
.
x
'
$+∞
I1 =
1
dx
.
x
+" &
$+∞
I1 =
1
dx
= ln x +∞
= ln(+∞) = +∞.
1
x
x
& & a arctg
> x1 ,
x
'
&, &" -,
#$. m
F
=
m
x2
M Ox
x
A
M x = a
$+∞
mdx
m
m
A=
− 2 = +∞
=− .
x
x a
a
a
! " ! ##$ %
&''
$1
I=
0
dx
√ .
x
( x → 0 #" % √1x
" ) # [; 1].
" " ##$ %
!!
$1
I = lim
→0
√
√
dx
√ = 2 lim x 1 = 2(1 − lim ) = 2.
→0
→0
x
* " $ !
&'+,
$2
I=
0
dx
.
x−1
( "
-! ! "
$1
I = I1 + I2 =
0
dx
+
x−1
$2
1
dx
.
x−1
I1 :
$1−
I1 = lim
→0
0
dx
= lim ln |x − 1| 1−
= lim ln = −∞.
0
→0
→0
x−1
I1
I2 !
+∞
"#$$
1
√
n n
n=1
% & ' ( "#" Un = f (n) = n√1 n
)* !
$+∞
I=
1
dx
√ =
x x
,"#-.
$+∞
1
$+∞
1
α = 3/2 +
dx
x3/2
dx
x2
/&
0! !
"#$-
"#$1
+∞
#
sin xdx.
0
$+∞
0
"#$"
$+∞
0
+∞
n=1
1
√ .
n n
"#$2
$2
√
0
dx
.
2−x
dx
.
a2 + b2 x2
xdx
.
c2 + x2
"#$3
$2
1
dx
.
x ln x
$27
−8
$1
0
dx
√
.
3
x
$+∞
−∞
dx
.
x2 + 2x + 2
arcsin x dx
√
.
1 − x2
$+∞
1
x3 + 1
dx.
x4
+∞
# √
xe−x dx.
0
$1
3
0
$1
x2 dx
.
(1 − x2 )5
dx
.
e −1
√
0
x
1
1
1
1
+
+
+ ··· ,+
+ ··· .
2 ln 2 3 ln 3 4 ln 4
n ln n
!" # $! %&
' ()
*+
, ' "
-"
#b
f (x) dx . / &
a
! f (x) 0 [a; b] 1 ' " - &
'0 F (x) / 2+ &3 "'! 4 5
$b
f (x) dx = F (b) − F (a).
a
. 1 '0 1 ' " / ! f (x)
0 /) ' )
) &
' + ' 1 / ) 1 '
' % "
!"
" ! #
$ % & f (x)
' (
) & "
! "
)
& "
% "
f (x) 0 x ∈ [a; b].
*+, -.
/! [a; b] n $
x0 = a < x1 < x2 < . . . < xn = b.
0 h h = b−a
n
! % !
y = f (x) $ A0, A1, A2 , . . . , Ai−1, Ai, . . . ,
An−1 An
y
y=f(x)
h
y0 y1
0
a
x1 x2
yn
b x
xn-1 xn
1 y = f (x) " 2 3
! n
$b
f (x)dx ≈ y1 h + y2 h + · · · yn h = h
n
yi .
i=1
a
y = f (x)
!
$b
f (x)dx ≈ y0 h + y1 h + · · · yn−1 h = h
yi .
i=0
a
"#
n−1
# !
$%
%# % %
% #&
y = f (x)
'(% )*
+ [xi−1; xi ]
(% +
& Ai−1 Ai ,
ΔSi =
yi−1 + yi
· Δxi .
2
Ai
y=f(x)
Ai-1
x
xi-1
-% (%
xi
+
(%
A0 A1 A2 · · · An .% .
+
f (x)
#b
a
f (x) dx
' %% (% ! #
*
(% #& +
#&
& )
! " # $
# h
%
Sn =
y1 + y2
yi−1 + yi
yn−1 + yn
y0 + y1
h+
h+ ...+
h+ ...+
h.
2
2
2
2
& '#
Sn = h
n−1
y0 + yn
+
yi .
2
i=1
(# & # ## & )'* + #''
)
$b
f (x) dx ≈ h
a
y0 + yn
+
yi
2
i=1
n−1
#'* + #'
,
h = b −n a ,
,-.
# ) n = 2m $ #)#
#* ' '' # ,,
Mср
y=f(x)
Mп
y=Ax2+Bx+C
Mл
yл
yср
yп
xл
x ср
xп
x
/& & # x x
" x
' f (x) # '* M M M
y = Ax2 + Bx + C.
$x
x − x
Ax2 + Bx + C dx =
(y + 4y + y ).
6
x
!
" #
Sn = (y0 + 4y1 + y2 )
b−a
b−a
+ (y1 + 4y2 + y3 )
+ ...+
6m
6m
b−a
+(ym−2 + 4y2m−1 + y2m )
=
6m
m−1
m
b−a
=
(y0 + y2m ) + 2
y2i + 4
y2i−1 .
6m
i=1
i=1
$
$b
h
f (x) dx ≈
3
y0 + y2m + 2
m−1
y2i + 4
i=1
a
h = b−a
!
2m
)
&'!,
$b
f (x) dx = lim
λ→0
(n→+∞)
a
m
y2i−1
,
%&'!&(
i=1
* + !
n
f (ξi )Δxi.
i=1
$b
f (x) dx ≈
a
n
f (ξi )Δxi ≈ h
i=1
h = Δxi = b −n a .
n
i=1
f (ξi ),
%&'!-(
ξi .
!"
y
y=f(x)
y0
h
0
x
# $ " " " %
&' ()* + , , "
$h
|y − y0 | dx.
δh =
0
- . ,
$
/
y = y0 + y (ξ)x.
-$
$h
δh =
0
|y (ξ|x dx
M1 h2
,
2
M 0 * ) y
[a; b].
1
1
= f (x)
n=
b−a
,
h
n
M1 nh2
,
δ δh · n
2
!
δ
#
b−a
M1 h.
2
$ !
&
% %
b−a
M2 h2 ,
12
b−a
M4 h4 .
δ
180
Mk = max |f (k) (x)|
δ
) *+
+
,
&
"
!
- -
!.
'
(
[a; b].
! ! % %/
0
b
$
f (x) dx − I2n |I2n − In | .
15
((
a
1 !
-0 *+
2 !
2 0 ! !
% -
! /
%
n
2n
! /
#1
(
I = 3x2dx
0
! "# $ %
& h = n1 = 0, 1 '" &( #
3
4 % .
$ !0
I=
#1
0
%
3x2 dx = x3 |10 = 1 & 0
! 0 .
. 5
I = 0,3(0,01 +
+ 0,04 + 0,09 + 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81 + 1) = 1,155
I = 0,3(0 + 0,01 + 0,04 + 0,09 + 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81) =
0+1
I = 0,3(
+ 0,01 + 0,04 + 0,09 +
= 0,855
2
+ 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81 + 1) = 1,005
!
I = 0,1(0 + 1 + 2(0,04 + 0,16 + 0,36 + 0,64) + 4(0,01 + 0,09 + 0,025 +
+ 0,049 + 0,081)) = 1.
" !
!# $ %
f (x) = 3x2 f (x) = 6x f (x) = 6 f (x) = f IV (x) = 0
[0,1]
M1 = 6 M2 = 6 M4 = 0 #
& '(
)* !
#
%
#+
b−a
1
M1 h = · 6 · 0,1 = 0,3,
2
2
b−a
1
M2 h2 =
· 6 · 0,01 = 0,005,
- δ
12
12
b−a
M4 h4 = 0
( δ
180
&
.
δ
0,155 < 0,3
#
)*
0,145 < 0,3
# )*
# !# )*
) %
! #
#
%
0 !#
/
1 2
! #
# # #
1
! ! !"!#
! $ 3 %
& ' % " "
( & #
" ! )*+ " #
#1
" ! I = cos x dx ' δ 0,0001?
/
/
0
3 ) + 4 % # 567(
*) ! $ $+
M1 = M2 = M4 = 1,
# %
!*
# )* )
10−4
! # !
h 0,0002 =⇒ n
1
= 5000.
0,0002
1
0,0012 =⇒ n √
≈ 29.
0,0012
h
h
4
0,0180 =⇒ n √
4
1
≈ 2.
0,0180
! " #$% &$" "% &
'$ " ( " )
"*+ $" , " -
&$" "% " ' ./
.
"*+ 0#
$b
+∞
a
n=0
f (n) (a) (x − a)n+1 b
f (n) (a)
(x − a)n dx =
=
n!
n!
n+1
a
n=0
+∞
=
δ = 0,005
+∞
f (n) (a)
(b − a)n+1 .
(n
+
1)!
n=0
Φ(x) =
#x
0
2
e−x dx
1 0 2 ! 3 ' 4 5 1'$
" * ( * " 6 2
$x
−x2
e
0
$x
dx =
0
(1 − x2 +
x4 x6 x8
x2n
−
+
+ . . . + (−1)n
+ . . .) dx.
2!
3!
4!
n!
7 $" &" -,
x = 1 .2
x4 x6 x8
−
+
+ · · · dx =
e
dx =
1−x +
2
6
24
0
0
1
1
1 x3
1
1
x9
1 1
1 x5 1 x7
= x − 0 +
−· · · .
+
−· · · = 1− + − +
0−
3
10
42 0 216
3 10 42 216
0
$1
$1
−x2
2
0
! 2161 < 0,005 0,005
I =1−
1
1
1
78
+
−
=
.
3 10 42
105
"#$
"#%
"#&
"#'
$
$
$
$
x dx
.
3x2 − 11x + 2
x3 + 5
dx.
x3 − x2 + 4x − 4
"#" tg5 x dx.
"#(
√
√
1
arcsin x
√
dx.
x+1
√
$
$4
3x2 + 1
dx.
+x+8
x3
x
x+1
dx.
"#)
+∞
n=2
ln n
.
n(1 + ln4 n)
x = −2 y = −x y = x12 .
R = 10
$
$
(3x2 + 1)dx
d(x3 + x + 8)
=
=
3
x +x+8
x3 + x + 8
$
= d ln |x3 + x + 8| = ln |x3 + x + 8| + C.
!
"# #
# $# %&''(
$
1
u = arcsin x, du = √1−x
arcsin x
2 dx
√
√
√
dx =
=
dx
√
dv = x+1 = d(2 x + 1) → v = 2 x + 1
x+1
$ √
√
x+1
√
dx = 2 x + 1 arcsin x−
2
1−x
√
√
= 2 x + 1 arcsin x + 4 1 − x + C.
√
= 2 x + 1 arcsin x − 2
$
√
−2
dx
1−x
'
) # # *
+ ,-$
' &.' *
t = 12 (3x2 − 11x + 2) = 3x − 11
/ +
2
11
z = 3t = x − 11
/
x
=
z
+
/
dx
=
dz
"/
6
6
,-$ 3x2 − 11x + 2 = 3(x2 − 11
x
+ 23 ) =
3
121
= 3 z2 −
97
= 3 (x − 11
x z + 116 dx
)2 + 23 − 36
6
36
dz
$
$
$
xdx
1
zdz
dz
11
√
=√
+ √
=
3x2 − 11x + 2
3
z 2 − 97/36 6 3
z 2 − 97/36
11
1 2
z − 97/36 + √ ln |z + z 2 − 97/36| + C =
=√
3
6 3
1
2 11
11
2 2
11
11
2
2
=√
ln x −
+ x − x + + + C.
x − x+ +
3
3
6
6
3
3 3
3
! "#
x3 −x2 +4x−4 =
= x2 (x − 1) + 4(x − 1) = (x − 1)(x2 + 4) $
$! %
$
x3 + 5
dx =
x3 − x2 + 4x − 4
$
1+
x2 − 4x + 9
(x − 1)(x2 + 4)
dx.
x2 − 4x + 9
A
Bx + C
=
+ 2
=
(x − 1)(x2 + 4)
x−1
x +4
=
A(x2 + 4) + (x − 1)(Bx + C)
⇒
(x − 1)(x2 + 4)
⇒ A(x2 + 4) + (x − 1)(Bx + C) = x2 − 4x + 9.
x = 1
5A = 6 → A = 6/5
x = 0 4A − C = 9 → C = 4A − 9 = −21/5 & $ !
$ '(( ! x2
A + B = 1 → B = 1 − A = − 15
)$ " $
$
3
x +5
dx =
x3 − x2 + 4x − 4
=x+
$
1+
6
1 x + 21
−
5(x − 1) 5 x2 + 4
dx =
6
1
x
21
ln |x − 1| −
ln(x2 + 4) −
arctg + C.
5
10
10
2
tg x
dz
tg x = z! x = arctg z! dx = 1+z
! "
$
$
5
tg xdx =
5
z
dz =
z2 + 1
2
z4 z2
z
dz =
− +
z −z+ 2
z +1
4
2
$
3
tg4 x tg2 x 1
1
−
+ ln(tg2 x + 1) + C =
+ ln(z 2 + 1) + C =
2
4
2
2
tg4 x tg2 x
=
−
− ln | cos x| + C,
4
2
# " ! " ln(tg2 x + 1) = ln cos12 x =
= −2 ln | cos x|
$ %
#
1
1
1
1
xm (a + bxn )p = x1/2 (x1/2 + 1)− 2 → m = ; n = ; p = − , a = b = 1.
2
2
2
" %
m+1
n
=
1
+1
2
√
√
x + 1 = z 2 → x = z 2 − 1,
√
dx
√ = 2zdz, xdx = 4xzdz = 4z(z 2 − 1)2 dz.
2 x
&'" ( x
!
$4
√
√
1
x
x+1
√
=4 3
"
√
dx = 4
$3
√
2
4
2
z − 2z + 1 dz = 4
=3
1/2
√3
z5 2 3
− z + z √ =
5
3
2
√
√
√ 4 4
16 3 28 2
9
−2+1 −4 2
− +1 =
−
=
5
5 3
5
15
√
√
7 2
4
4 3−
≈ 2,903.
=
5
3
$ )
$+∞
2
$+∞
ln x
dx =
x(1 + ln4 x)
2
1
ln x
d ln x =
4
2
1 + ln x
$+∞
2
d ln2 x
=
1 + ln4 x
+∞
1 π
1
2
− arctg ln2 2 .
=
= arctg ln x
2
2 2
2
+∞
ln n
! " ! "!
#
n(1 + ln4 n)
n=2
$ ! "
%&'
! (
)*+ %, - . ( '/ ! ! #
! ( xB = −1 ! (0 1!
y=1
-x
y=
C
/x 2
y
B
A
-2
-1
$−1
−x −
−2
1
x2
dx =
x
SABC =
0
−
−1
x2 1
4 1
1
+
= − − 1 + + = 1.
2
x −2
2
2 2
%&
! 2
%' 3 " !" 0 " ! Δx
( x
2
x + y 2 = R2
ΔA = gxΔm = gxρΔV ≈ γxπy Δx = 2
y = R 2 − x2
2
=
-R
0
R
y
x
x+ Δ x
x
R
= πγx(R2 − x2 )Δx.
R
x4
−
A = πγ(R x − x )dx = πγ R
=
2
4 0
0
πγR4
π
R4 R4
−
=
= 107 ≈ 7854000
= πγ
2
4
4
4
γ = 1 3 = 103 3
$R
2
3
2x
2
.
$16
√
1
$
$
dx
.
3 cos2 x + 4 sin2 x
(x + 1)ex dx.
$
4 − 3x
dx.
2
5x + 6x + 18
$
dx
.
−1
x4
$
sin2 x
dx.
cos4 x
dx
√ .
x(1 + 4 x)3
r = 2(1 − sin ϕ).
+∞
n
.
3n2
n=1
R H !
γ
S =
ΔSi
n
ΔSi
i=1
λi ΔSi
ΔSi
λ
max λi .
ΔSi λ = i=1,2,···
,n
! " # λ → 0 "# λi → 0
$ " % "
Pi ∈ ΔSi ξi , ηi &# '( $ # f (Pi ) =
f (ξi ; ηi ) ! " f (ξi ; ηi )ΔSi ## " #
n
f (ξi ; ηi )ΔSi "# ΔSi .
i=1
f (ξi; ηi)ΔSi n
i=1
z = f (x, y)
S
* f (x, y)
S
n
n
f (ξi; ηi)ΔSi ! ! !
i=1
" ΔSi ! λ
#
n
)
+
#,
##
f (x, y)ds "#
S
##
, # S ΔSi "
Pi ∈ ΔSi -
$$
n
lim
f
(ξ
;
η
)ΔS
=
f (x, y)ds.
i i
i
n→+∞,
λ→0
i=1
& (
S
##
.#
" #, " S / 0
# " , f (x, y) / 1 ds / 0
f (x, y)ds / " ! "0
n
" # #", ,
ΔSi = S #"
i=1
lim
n→+∞,
λ→0
n
i=1
$$
ΔSi =
ds = S,
S
& )(
S
S
##
z = f (x, y) f (x, y)ds
S
! " # z =
= f (x, y) $
S
% $ & " # '( z = f (x, y)
) z = f (x, y) 0
S * !
σ
!
)
z = f (x, y) " Oxy S & +,(
- ! Pi ∈ ΔSi " ! Mi ∈ σ
Oxy
z
Mi σ
ζi
ηi
y
0
ξi
Pi S
x
. ζi = f (ξi ; ηi ) = Pi Mi f (ξi ; ηi )ΔSi = ζi ΔSi
/0 " ΔSi $ Pi Mi = ζi
n
n
1
f (ξi ; ηi )ΔSi =
ζi ΔSi /0
n=1
n=1
2 ! 3 Vn
) "
lim
n→+∞
n
$$
f (ξi ; ηi )ΔSi =
i=1
f (x, y)ds = lim Vn = V,
n→+∞
S
V
Oz S
σ ! "#$
%$ f (x, y) 0 S
&$ !! ' $ #
$" (
) # (' & )
!
• &$' ' !
*
$$
$$
kf (x, y)ds = k
f (x, y)ds,
S
•
S
+ ' !%
!% *
$$
$$
(f (x, y) + ϕ(x, y))ds =
S
•
'
$$
f (x, y)ds +
S
• ,
## )
f (x, y)ds 0
•
k = 0.
ϕ(x, y)ds.
S
$ f (x, y) 0
S
, ) $ m M $ $"$
- ) - ###$ !%
f (x, y) m f (x, y) M mS
f (x, y)ds MS
S
. ( %! m
, ) $ S = Sj
j=1
$$
f (x, y)ds =
S
m $$
j=1 S
i
f (x, y)ds.
•
f (x, y)
S S
##
P (ξ; η)
f (ξ, η) = S1
f (x, y)ds
S
f (x; y; z) S.
! "! #
!
f (x, y) 0 !
$ f (x, y) 0 S
%
%
S
y = y (x) ! & y = y (x)
x = a x = b ' ()* %
x = const
Oy a < x < b !
" $ y = y (x)
!
" !
y = y (x) + () , $
-./ -0/
y
x=const
y=yB (x)
K
B
A
S
D
C
E
a
b
y=yH(x)
x
Oy
1) 2 S
a < x < b y y = y (x) y = y(x)
OY x = const ! ! "
Oy
A D B C
y = y (x) y = y(x)
f (x, y) 0 S
f (x, y) S ! "
S σ
#$
z
z=f(x,y)
σ A’
K’
D’
E’
C’
B’
D
a
y
A
K
E
S
b
C
x
B
x =
% & ' x = ()*+, "
EE K K -. #$ -
#/
y #(x)
01 & Q(x) =
f (x, y)dy & x
y (x)
2 ! ABCDAB C D - "
1' 34# 5
$b
V =
$b
Q(x) dx =
a
a
⎛
⎜
⎝
y$ (x)
⎞
⎟
f (x, y)dy ⎠ dx.
y (x)
z
z=f(x,y)
E’
K’
y=yB (x)
Q(x)
y=yH (x)
0
E
y
K
x =
$b
y$ (x)
dx
V =
f (x, y)dy.
y (x)
a
f (x, y) 0 :
$$
f (x, y)ds.
V =
S
V
y x
$$
$b
f (x, y)ds =
y$ (x)
dx
a
S
f (x, y)dy.
!
y (x)
" # $
# $ % & '(
$ x = )*+,- y = )*+,- & S
& $ ./!
" 0 1 ! 2
$$
$b
f (x, y)ds =
S
$d
dx
a
f (x, y)dy.
c
/!
y
d
S
c
0
a
dxdy
= ds
x
b
f (x, y)
(x,y)
x = const F (x, y) ∂F∂y
= f (x, y)
y
x
!
"#
$b
$(x) (x)
$b
y
dx
f (x, y)dy = F (x, y) dx =
y
y (x)
a
a
y
"
$b
(F (x, y (x)) − F (x, y (x))) dx.
=
a
$ % & '
(
x & )
%
) S !(
) !
S
) (
% % Oy %
!
Si ) )
* %
+
S ' a x c
y = y2 (x)
y = y1 (x) c x b
y
y=yB (x)
y=yB (x)
2
1
S1 S2
S3
y=yH (x)
y=yH (x)
2
1
a
0
c d
b
x
a x d y = y 1 (x)
d x b y = y 2 (x) S
S1 , S2 , S3
$$
$$
f (x, y)ds =
S
$$
f (x, y)ds +
S1
$c
y$1 (x)
dx
=
a
y1 (x)
$$
f (x, y)ds +
f (x, y)ds =
S2
$d
f (x, y)dy +
S3
y$1 (x)
dx
c
y2 (x)
$b
f (x, y)dy +
y$2 (x)
dx
f (x, y)dy.
y2 (x)
d
##
S
O(0; 0) A(1; 1) B(1; 0)
(x + y)ds
S
!
" y = 0 x = 1 y = x # x = const 0 < x < 1
$ % &
'% ()*
% + y = y (x) = 0
'% (,* % !
+ y = y (x) = x -
S ' '
x " . / 001
y
A(1;1)
x=const
y=x
O(0;0)
y=0
B(1;0) x
$$
$1
(x + y)ds =
0
S
$1
=
0
$x
dx
0
3
(x + x /2)dx =
2
2
2
$1
(x + y)dy =
x
(xy + y 2 /2) dx =
0
0
$1
x2 dx =
0
1
3 x3
1
= .
2 3 0 2
# dx # (x2 + 3y2)dy.
2
0
1
0
! "#
#1
(x2 + 3y2)dy x
0
y $ % & '% x
$2
$1
dx
0
(x2 + 3y 2 )dy =
0
$2
=
1
(x2 y + y 3 ) dx =
0
0
2
14
x3
8
+ x = + 2 = .
3
3
3
0
(x2 + 1)dx =
0
$2
$2
$x
dx
1
x2 dy
.
y2
1/x
! x = "#$%& x2 ' ' ( (
( ! ! ! '
$2
x2 dx
1
$x
dy
=
y2
1/x
$2
=
$2
x2 −
1
(x3 − x)dx =
1
x
$2
1
1
2
dx =
dx
=
x
x
−
y 1/x
x
1
2
9
x4 x2
1 1
−
=4−2− + = .
4
2 1
4 2
4
' ) * '
! S (' +
)
#3
1
dx
x+9
#
f (x, y)dy.
x2
,
x+9
#
x2
f (x, y)dy !
! x = const -#. ! / y = x2
01 ! S ( +
! -2. ! y = x + 9 01 S 3+
4' ! S x !0 ! !' x = 1
x = 3 5 ! !
x=1
y
10
y=x+9
B
C
x=const
A
2
1
D
0
1
x=3
y=x
x
3
y = x2 y = x + 9 x = 1 x = 3
S
y = x+9
x=1
x=3
! "# $ %! &# '
y
y= 25-x
2
y=- 25-x
+ ,
#3
−4
√
dx
25−x
# 2
√
− 25−x2
x
x=3
x=-4
x=const
( ! # ) %!*#
f (x; y)dy.
2
y = x2
√
25−x
# 2
√
f (x; y)dy
− 25−x
x = const √
y = − 25 − x2 → x2 + y2 = 25!
x "
√S #! $
% & y = 25 − x2 → x2 + y2 = 25!
" S ' ( )*
S x * x = −4 x = 3' + ! $
x2 +y2 = 25 *$
x = −4 x = 3' , # #
S ' -.
/ x2 + y2 = 25 * x = −4
x = 3 * A(−4; 3), B(3; 4), C(3; −4) D(−4; −3)'
2
$4
01'0
$y
dy
2
01'-
0
$4
$2
dx
3
01'2
$2
1
x2
y3
dx
+ y2
dy
(x + y)2
$2x
dx (2x−y)dy
301'.4 301'54 *!
" # # & '
1
01'.
01'5
√
$2
$ 4x
dx
0
$1
0
x
√
2x−x2
f (x, y)dy
√
2
$3−y
dy
f (x, y)dx
y/2
$
" # $
##
%&'
(x + y)ds S
S
O(0; 0), A(1; 1), B(0; 2)
!
y B(0;2)
x=const
y=2-x
A(1;1)
y=x
x
O(0;0)
( ) * +$ ,
x = 0, y = x, y = 2 − x -.
! " / ) %''
2−x
##
#1
#
#1
# $" (x+y)ds = dx (x+y)dy = (xy+
+y
2
/2)|2−x
x dx
− x2 −
x2
)dx
2
=
#1
0
0
S
2
2
(x(2−x)+ (2−x)
−x2 − x2 )dx
2
#1
= 2 (1 − x2 )dx = 2(x −
0
x3 1
)|
3 0
#1
0
x
2
2
= (2x−x +2−2x+ x2 −
0
= 2(1 − 13 ) = 43 .
##
S
O(0; 0) A(1; 1) B(2; 0)
(x + y)ds
S
x = 0 y = x y = 2 − x
y
x=const x=const
A(1;1)
y=2-x
y=x
O(0;0)
C(1;0)
B(2;0)
x
OY ## OAC
ACB !
(x + y)ds =
=
##
(x + y)ds +
##
S
(x + y)ds
"
OAC ACB #$
2−x
##
#1 #x
#2
#
(x + y)ds = dx (x + y)dy + dx (x + y)dy =
OAC
= (xy +
0
=
0
S
#1
+
ACB
y2 x
)| dx
2 0
#1
#2
+ (xy +
1
(2−x)2
)dx
2
=
1
2
−2+
+4−
4
3
3
2
0
2
#2
y 2 2−x
)| dx
2 0
x dx + (2 −
1
6
= 43 .
1
x2
)dx
2
0
#1
1
2
= (x +
0
=
3 x3 1
|
2 3 0
x2
)dx
2
#2
0
+ (x(2 − x) +
+ (2x −
1
x3 2
)|
6 1
=
S x = x (y)
x = x(y) y = c y = d
y
A
d
B
x=x (y)
x=xnp(y)
y=const
c
D
C
x
Ox
y = const Ox
! !
"
#
x = x (y) ! $ !
" # x = x(y) %
& $ ' ()* (+*
,- S
c < y < d x (x = x(y) x = x(y))
OX y = const
Ox
. A B ' D C !
$ x = x (y) x = x(y)
/ $
x ' 0
0 , 1
!$!2 # ! 3
y4
c < x < d
$$
$d
f (x, y)ds =
S
x$ (y)
dy
c
f (x, y)dx.
x (y)
,-
$$
$d
f (x, y)ds =
$b
dy
c
S
f (x, y)dx.
a
! !" " ! #
$" % & ' (
f (x, y) = ϕ(x)ψ(y) ! & !
! )
!*") +
$$
$b
f (x, y)dxdy =
$d
ϕ(x)dx
a
S
ψ(y)dy.
,
c
' " ! & ! f (x, y) y = -./01
y)
= f (x, y) (
$2 Φ(x, y) ! ∂Φ(x,
∂x
x ! #
% +
$d
x$ (y)
dy
c
$d
f (x, y)dx =
x (y)
c
x
Φ(x, y)
(y)
x (y)
dy =
$d
(Φ(x (y), y) − Φ(x(y), y)) dy,
=
c
! ! 3 ! !*#
y
,
4 3 + 5 ! y = -./01 0 < y < 1
2 6.7 #
x = x(y) = y 2 687
x = x (y) = 1 9 & ) S & y
" : % ;
y
A(1;1)
x=y
y=const
B(1;0) x
O(0;0)
$$
$1
(x + y)ds =
$1
dy
0
S
$1
=
0
$1
(x + y)dx =
0
y
1
3
+ y − y 2 dy =
2
2
y y2 y3
+
−
2
2
2
x2
1
+ yx dy =
2
y
1
1 1 1 1
= + − = .
2 2 2
2
0
y = 0 < y < 1 !
"# $ % & ' ( ) x = x (y) = 0 #
&# # & 0 < y < 1 ' *+ & '
& x = x(y) = y 1 < y < 2 , x = x(y) = 2 − y
& S & -
'
# OAC
##
## CAB .& /01
##
2# - (x + y)ds = (x + y)ds + (x + y)ds.
S
OAC
CAB
&& 3 ' '$ ' 4
# OAC CAB && & & /0 - '4
"
$$
$1
(x + y)ds =
S
$y
dy
0
$2
(x + y)dx +
0
1
$2−y
4
dy (x + y)dx = .
3
0
y B(0;2)
x=2-y
y=const
A(1;1)
C(0;1)
y=const
x=y
x
O(0;0)
##
S
2−y
#1
#
(x + y)ds = dy (x + y)dx = 43 .
0
y
y
A(1;1)
y=const
x=2-y
x=y
O(0;0)
C(1;0)
B(2;0)
x
!! " # $
$% $ $ &'
% $% $ $%
$( $ % $% $% $(
$ $%
f (x, y)
S
a x b, c y d !"# $
S x = %&'() y = %&'() *
+ [a; b]
k
m
k [c; d] m , b − a = Δxl c − d = Δyj
j=1
l=1
Δxl = xl − xl−1 Δyj = yj − yj−1
y
ym =d
yj
yj-1
y0 =c
x0=a
xl-1 xl
x k =b x
+
n -
S = (b − a)(c − d) n = km
ΔSi = Δxl · Δyj
f (x, y) (ξl; ηj )
. !"
f (ξl; ηj )ΔxlΔyj .
+ / j 0 *
1 !"# m−
m
j=1
f (ξl ; ηl )Δxl Δyj ,
y
l
0 Δyj → 0
Δxl →
lim
k
Δxl →0
Δyj →0
=
#b #d
a
l=1
m
f (ξl ; ηj )Δxl Δyj
j=1
= lim
Δxl →0
Δyj →0
k
m
l=1
j=1
f (ξl ; ηj )Δyj
x
Δxl =
##
#b #d
f (x, y)dy dx = dx f (x, y)dy =
f (x, y)ds.
c
a
c
S
ABCD
oy ! " #
y
y=yB (x)
A
B
D
y=y H (x) C
yj
yj-1
0
x0=a
x k =b x
xl-1 xl
Oy
$
%
&
, - % y = y (x)
y = y (x) $
Δxl Δyj , &
% , , x = '()*+
,
# .
/(0 . /10
y y = y (x) y = y (x)
% 2
%
l Δxl , 3
4!5# 6
3 47!#
Ox "!#
x = '()*+ y = '()*+ %
! "
#
$3
1
$x+9
dx
f (x, y)dy.
x2
$ % & '( !
( )* ** +
( ,
x ( (
y = 9 y = 10 - ./
01 ( ** 2 3 (,
456./7.01/7081
$$
$$
$$
$$
f (x, y)ds =
S
f (x, y)ds +
DEC
f (x, y)ds +
EAKC
f (x, y)ds.
ABK
$ &
2 '( 6./ x = 1, x = √y,
x,
9 - **2
: , ;( y = ? 0
∂x
∂y
%
δ !
" δ
P dx + Qdy =
C
∂Q
∂x
## ∂Q
∂x
δ
−
∂P
∂y
## ∂Q
δ
−
∂P
∂y
∂x
> 0
dδ > 0
−
∂P
∂y
%
dδ " C #
δ
P dx + Qdy > 0
C
$ %
" "
D
∂P
∂Q
=
.
∂y
∂x
&'(
$
ydx + xdy
AB
) * + , "
#
$
ydx − xdy.
AB
ydx+xdy :
P = y Q = x
AB
∂P
∂y
= 1
" "
% %
&'- &'.
#
/ "
ydx − xdy +
P = y Q = −x ∂P
= 1 ∂Q
= −1
∂y
∂x
∂Q
∂x
=1⇒
∂P
∂y
=
∂Q
∂x
AB
∂Q
∂P
=
"
⇒
∂y
∂x
"
%
&'- &'.
P dx + Qdy
P = P (x; y) Q = Q(x; y)
D 0
$
U = U (x; y) %
&'1
!
"
P dx + Qdy
D
U = U (x; y)
D
∂P
∂Q
=
.
∂y
∂x
P dx+Qdy
U = U (x; y)
∂U
∂U
dx +
dy = P dx + Qdy.
dU =
∂x
∂y
! " ∂U
= P ∂U
= Q # "" "
∂x
∂y
∂ U
$ y " x % ∂x∂y
= ∂P
& ∂ U = ∂Q
'
∂y ∂y∂x
∂x
∂Q
∂P
∂x " " (" ()( $ *
∂y
" U (x; y) " " ' *
* % ∂P
= ∂Q
∂y
∂x
+ ! U = U (x; y)
2
2
" # $ P dx + Qdy
% & #
#
, % P dx+Qdy *" " " (
)- % A B *" $ % A B
(x;y)
(x;y)
#
#
* % x0 y0 x y '.
dU (x; y) =
P (ξ; η)dξ +
(x ;y )
(x ;y )
+ Q(ξ; η)dη "
AB
0
0
0
0
(x;y)
$
U (x; y) =
P (ξ; η)dξ + Q(ξ; η)dη + C.
(x0 ;y0 )
/0
1 ( . % x y *"
% ." ( * %
" ξ η 2%" % ( U (x; y) (( % )
C = U (x0; y0)
3 ." " /0 " " )
()- % A(x0; y0) B(x; y) 4
* ." ( ")-
ABC *" ( AB CD (" ()-(
η = y0 = ξ = x =
dη = 0 dξ = 0
$x
U (x; y) =
$y
P (ξ; y0 )dξ +
x0
Q(x; η)dη + C.
y0
y
y
B
y0
C
A
x0
x
x
U = U (x; y)
P (x; y) Q(x; y) !
"# ! $ % &' x0 = y0 = 0
$x
U (x; y) =
$y
P (ξ; 0)dξ +
0
Q(x; η)dη + C.
(
0
º
ydx + xdy
! " ydx + xdy ##$
% & & # ' ()*
% $ % y "
$x
U (x; y) =
$y
0dξ +
0
xdη = xy + C.
0
+ dU , dU = d(xy+C) = ydx+xdy
1
x
y
dx + y2 − yx2 dy
+
1
y
P = x1 + y1 Q = y2 − yx2
1 ∂Q
1
∂P
∂P
=
−
=
−
= ∂Q
∂y
y 2 ∂x
y2
∂y
∂x
!"# $# % &'
% ( ) **# *+# $ $ $
x = 0 y = 0 )$, P (x; y) Q(x; y)
) ** x0 = y0 = 1
$x
$y
1
2
x
+ 1 dξ +
− 2 dη =
U (x; y) =
ξ
η η
1
1
y
x
=
= (ln ξ + ξ)|x1 + 2 ln η +
η 1
x
x
= ln x + x − ln 1 − 1 + 2 ln y + − 2 ln 1 − x = ln x + 2 ln y + + C.
y
y
1 1
2
x
∂U
∂U
= + ./0#1
= − 2 = Q(x, y)
$ -
∂x
x y
∂y
y y
' % ! ' $
-' ' ! %3 -
#
" !
AB
*
2xydx − x2 dy O(0; 0) A(2; 1) !
O
A # $$%&'
• - OaA
2
1
4 - OaA y = x ⇒
2
1
dy = dx
2
2
$
$2
1
4
1
2
2xydx − x dy =
x2 − x2 dx = x3 = .
2
6 0 3
OaA
0
y
A(2;1)
C(0;1)
c
y=Const
a
b
O(0;0)
• ObA
Oy
x
dy = dx
2
$
2xydx − x2 dy =
$2
0
ObA
x
B(2;0)
x 3 x3
−
2
2
• OcA Ox
ObA y =
x2
⇒
4
dx = 0.
OcA y =
x
⇒
2
dx
dy = √
2 2x
$
2
$2
2xydx − x dy =
OcA
0
√ $2
√
1 √
3 2
x 2x − x 2x dx =
x3/2 dx =
4
4
0
2
√
12
3 2 2 5/2
· x = .
=
4
5
5
0
• OBA
OBA
OB BA OB − y = 0 ⇒ dy = 0 BA − x = 2dx = 0
$
$1
2
2xydx − x dy = 0 +
(−4)dy = −4.
0
OBA
• OCA
OCA
OC CA OC − x = 0 ⇒ dx = 0 CA − y = 1 ⇒ dy = 0
$
$2
2
2xydx − x dy =
OCA
#
2
2xdx + 0 = x2 0 = 4.
0
2xydx + x2 dy
OA
•
! OaA
$
2
$2
2xydx + x dy =
0
OaA
2
$2
1 2
x3
3
2
= 4.
x dx =
x + x dx =
2
2
2 0
2
0
•
" ObA
$
2
$2
2xdx + x dy =
0
ObA
x3 x3
+
2
2
$2
dx =
0
2
x4
x dx =
= 4.
4 0
3
•
" OcA
$
2
$2
2xdx + x dy =
0
OcA
=
√
1 √
x 2x + x 2x dx =
4
√ $2
√
5 2
5 2 2 5/2
· x = 4.
x3/2 dx =
4
4
5
0
•
$
OBA
2xdx + x2 dy = 0 +
$
$1
4dy = 4.
0
OBA
•
OCA
2xdx + x2 dy =
$2
2
2xdx + 0 = x2 0 = 4.
0
OCA
!
L
• L " OBAaO
P = 2xy Q = −x2
#
∂Q ∂P
−
= −2x − 2x = −4x,
∂x
∂y
&
2xydx − x dy = −4
L
$1
= −4
0
$1
$$
2
xdS = −4
S
2
$1
x2
dy = −2 (4 − 4y 2 )dy = 8
2 2y
0
$2
dy
0
xdx =
2y
1
y3
16
− y = − .
3
3
0
$ % &
% x y = '()*+ %
y = 2x x = 2
P = 2xy Q = x2 ⇒
= 2x
2xydx + x2 dy
L
! " # $ " %
"& $ L "# O(0; 0) A(2; 1) #
'
(
∂Q
#∂x
∂P
∂y
2xydx + x2dy
) # " !%
**+ , ) #
# # # # ! $ - ,%
, * $
.
$y
U (x; y) = 0 +
x2 dη = x2 y + C.
o
/ # $ 0
∂U
∂U
= 2xy 1
= x2 .
∂x
∂y
#
2
2
(x − 2xy)dx + (2xy + y )dy
AB y = x2
A(1; 1) B(2; 4)
#
(2a − y)dx + xdy C
C
x = a(t − sin t) y = a(1 − cos t) 0 t 2π
2 !" " #
%
2(x2 + y 2 )dx + (x + y)2 dy
C
C "
$ " % A(1; 1) B(2; 2) C(1; 3) &
" "
"
" " ABCA
.3 '
x = a cos3 t y = a sin3 t
AB
(3x2 − 2xy + y 2 )dx − (x2 − 2xy + 3y 2 )dy
ex−y ((1 + x + y)dx + (1 − x − y)dy)
dy
dx
+
x+y x+y
!
" # $ % & #
' ( ( #
)! ! !
&
* (
! " #
#
+
,# ( D Oxyz
#
F ! Fx = P = P (x; y; z)- Fy = Q = Q(x; y; z)
Fz = R = R(x; y; z)- !. # x, y, z
D !
/
F = F (x; y; z) = P (x; y; z)i + Q(x; y; z)j + R(x; y; z)k
0 /1
" # $ %
/2 & L !
F = (P ; Q; R)
' ! "
#
# ! ! C
/3 (
) !" " C " "
z
Δσi
ni
Pi
σ
ζi
ηi
x
y
ξi
σ
n D
n σ = Δσi Pi ξi ηi ζi
i=1
! Δσi "
# ni = n(ξi; ηiζi) # σ
$ n σ #
%! # & % Oz '& #
& $# Fi = F (ξi; ηi; ζi)
ni Fi # Δσi #
Pi ( n Fi · ni · Δσi
n)% & % *n = Fi · ni · Δσi
+,- n
n
i=1
n = Fi ·ni ·Δσi
Δσi
i=1
n → +∞ F
σ
$$
$$
σ = F · ndσ = F (x; y; z) · n(x; y; z)dσ.
+,.
σ
σ
/ n #
0 % σ
σ
$$
=
F · ndσ.
σ
!" #" $ D %
$&" $ W = W (x; y; z)
D " ' $ # #"
F = W (x; y; z)
()# # # ) " # # %
* +" &
" #" ) "
' , " # # %
& " #
+ " ' # # %
# - ' '
& # # #
(
) # )
$ σ # # + " * #%
$& γ
- ) # # " ) $ W
## " $& %
* Δσ ./0
W
Δσ1
n
h
ϕ
Δσ
Δσ
1 + # + " %
* Δσ " # * Δσ1 W
" ) 2 ΔQ"
Δσ
Δσ W
γ h ΔQ = γhΔσ
n ! Δσ ϕ ! "
W ΔQ = γW cos ϕ · Δσ = γ(W n)Δσ
# $ %&' ()
* * σ
$$
W (x; y; z) · n(x; y; z)dσ.
Q=γ
%&' +)
σ
&' &
F
= (P (x; y; z); Q(x; y; z); R(x; y; z)) M(x; y; z) ∈ ΔV
F σ
ΔV M
ΔV → 0
div
$$
1
F ndσ
ΔV →0 ΔV
div F = lim
σ
∂Q ∂R
∂P
+
+
∂x
∂y
∂z
$$
$$$
=
F · ndσ =
div F dV.
div F =
"σ
σ
!
#
V
$ % &'
( )* ) + )* )
#
! "# $$$%% & ' (
div F = 0
##
##
F ndσ +
σ1
σ1
σ2
F ndσ = 0.
σ2
n
σ2
F
n1
F
n
σ1
n1 = −n
##
F n1 dσ =
σ1
##
n
σ
F ndσ
σ2
div F = 0
! "
div F = 0
#
B
%
L
$
D #
n
A
$
n
F (ξi ; ηi ; ζi )Δli =
i=1
=
n
P (ξi ; ηi ; ζi )Δxi + Q(ξi ; ηi ; ζi )Δyi + R(ξi ; ηi ; ζi )Δzi .
i=1
#
n
i=1
" &
n
F (ξi ; ηi ; ζi )Δli
!
! !
AB
n → +∞
$
$
F (x; y; z)dl =
L
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz
L
$
$
F (x; y; z)dl =
AB
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz.
AB
z = z(t)
L
$
x = x(t) y = y(t)
P dx + Qdy + Rdz =
AB
$tB
=
(P (x(t); y(t); z(t))x(t) + Q(x(t); y(t); z(t))y (t)+
tA
+R(x(t); y(t); z(t))z (t)) dt,
tA
#
tB
!
"
A
B
%&%
F = F (x; y; z) = (Fx(x; y; z); Fy (x; y; z); Fz (x; y; z))
( ) *
A
t
$
$
B
-
+
$
F (x; y; z)dl =
A=
AB
'
( (,
'
L
Fx (x; y; z)dx + Fy (x; y; z)dy + Fz (x; y; z)dz.
AB
!
L " " F
L
&
&
F dl =
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz.
L
L
/ 0
L
1
. (" " 1%
)
"- "
1
.
- 3
( (&
4&%
%
'
(" 2 '
("
F = P (x; y; z)i + Q(x; y; z)j + R(x; y; z)k
rot F
∂R ∂Q
∂P
∂Q ∂P
∂R
−
−
−
rot F =
i+
j+
k =
∂y
∂z
∂z
∂x
∂x
∂y
i j k
∂ ∂ ∂
= ∂x ∂y ∂z .
P Q R
! " #
$" % & rot F ' !
$% ! ( ! (
! ( $" ∂x∂ ) ∂y∂ ) ∂z∂ & P * Q* R
+ $ ∂P
) ∂P ) ∂P ) ∂Q
∂x ∂y ∂z ∂x
* % ! & rot F = ∂R
− ∂Q
; ∂P − ∂R
; ∂Q − ∂P
∂y
∂z ∂z
∂x ∂x
∂y
%
,"$! rot F & "& F dl " "
L
* #$ $ !
- P = P (x; y; z)!
Q = Q(x; y; z)! R = R(x; y; z)! "#
F =
F (x; y; z)! $ σ !
L! F
L % & σ'
&
$$
F dl =
L
n rot F dσ.
.
σ
/ . $" ! $
" ' % % *
& "&" $ L " "
% & % ( *
. rot V $ + !*
L 0 " $" * + !
! "
P= P (x; y)# Q = Q(x; y)
k & $
R = 0 $ % rot F = ∂Q
− ∂P
∂x
∂y
L ' " σ ( ( Oxy n = k )
* + rot F n (
F = (P (x; y; z); Q(x; y; z); R(x; y; z))
rot F = 0
, U = U (x; y; z)
F = =F (x; y; z)
F = F (x; y; z)
- . /0
% ! F = F (x; y; z) =
"
P (x; y; z)i+Q(x; y; z)j +R(x; y; z)k
rot F =
∂R ∂Q ∂P
∂R ∂Q ∂P
−
;
−
;
−
∂y
∂z ∂z
∂x ∂x
∂y
= 0.
# $ $
$
∂R ∂Q
−
= 0;
∂y
∂z
∂P
∂R
−
= 0;
∂z
∂x
∂Q ∂P
−
= 0.
∂x
∂y
" '1. " " ( F = F (x; y; z) )
( 2 U = U (x; y; z) . " $ ( ./
, / $ . %3
F = grad U =
.
P =
∂U
,
∂x
∂U
∂U
∂U
i+
j+
k
∂x
∂y
∂z
Q=
∂U
,
∂y
R=
∂U
.
∂z
ξ η ζ
(x; y; z) C = U (x0 ; y0 ; z0 )
z
B(x;y;z)
A(x 0;y0 ;z 0)
C(x;y0 ;z0)
D(x;y;z 0)
y
x
L
ACDB AC CD
DB
!""#
η = y0 ζ = z0 $ ξ = x ζ = z0 $ ξ = x η = y dη = dζ = 0$ dξ = dζ = 0$
dξ = dη = 0 % & ' ()!*#
$x
U (x; y; z) =
$y
P (ξ; y0 ; z0 )dξ +
x0
$z
Q(x; η; z0 )dη +
y0
R(x; y; ζ)dζ + C.
z0
()"+#
x0 =
()! P Q R
y0 = z0 = 0
O(0; 0; 0)
()" ! " # $
# %& &' "' U = U (x; y) $
# ( " # %& )' "' U =
= U (x; y; z) *
" + + $
# " #
& , # - #
F = kr = kxi +
R
+kyj + kzk
x2 + y 2 + z 2 = R2
n = R1 R
F = kR F n = kR · R1 R = Rk R2 = kR
F = kr
x2 + y 2 + z 2 = R2
$$
$$
σ =
F ndσ = kR
dσ = kR4πR2 = 4πkR3 .
σ
σ
! " # " "
P = kx$ Q = ky $ R = kz % &
div F = ∂P
+ ∂Q
+ ∂R
= ∂kx
+ ∂ky
+ ∂kz
= 3k #
∂x
∂y
∂z
∂x
∂y
∂z
' " &( )
σ =
$$$
$$$
div F dV = 3k
V
4
dV = 3k · πR3 = 4πkR3 .
3
V
*
F = (2x + z)i + (y + 2z)j + (z − y)k
x = 0 y = 0 z = 0 x − 2y + 2z − 6 = 0
+ , - . / "
0 1 2*3 ' ! " F
# " " , P = 2x+z $ Q = y +2z $ R = z −y
div F =
∂(2x + z) ∂(y + 2z) ∂(z − y)
+
+
=2+1+1=4
∂x
∂y
∂z
)
σ =
$$$
$$$
div F dV = 4
V
dV =
V
= 4 · 61 OA · BO · OC = 4 16 · 6 · 3 · 3 = 36.
z
3
y
0
C
A
6 x
y=const
x=6+2y
x=6-2z
z=const
0
A
6 x
y = 0
•
3B
z = 0
ABC : x − 2y + 2z − 6 = 0
N = Ai + Bj + Ck = i − 2j + 2k ⇒ n =
N
=
N
i − 2j + 2k
2
2
1
=√
= i − j + k.
3
3
3
1+4+4
F n
F n = 13 (2x + z − 2y − 4z + 2z − 2y)
z = 3 − x2 + y
F n = 56 x − 52 y − 1
! "#$ %& dσ = 1 + z 2x + z2y dxdy =
= 1 + 14 + 1dxdy = 32 dxdy
'ABC = 23
$$
5
5
x − y − 1 dxdy =
6
3
AOB
3
=
2
3
=
2
=
3
2
$0
−3
#0
−3
$0
6+2y
$
dy
−3
0
5
5
x − y − 1 dx =
6
3
5
5
2
(6 + 2y) − (6 + 2y) − (6 + 2y) dy =
12
3
9 − 2y − 53 y 2 dy =
3
2
0
9y − y 2 − 53 y 3 −3 = 31,5.
OABC σ = BOC + AOC + AOB +
+ ABC = −4,5 + 18 − 9 + 31,5 = 36.
! " " #
$
"
% & '()#
*+#,
•
z = x2 + y2
x2 + y2 = 4 !"
F = xy 2 i +
yz
j
2
+ x2 zk
$ - .
/ -
z = f (x; y) 0 %'(#1()
−fx (x; y)i − fy (x; y)j + k
−2xi − 2yj + k
n=
.
=
1 + 4x2 + 4y 2
1 + f 2x (x; y) + f 2y (x; y)
/ n Ox
#
−2x2 y 2 − y 2 z + x2 z
F n =
#
1 + 4x2 + 4y 2
2 "
=
$$
σ
3
2 2
2
2
−2x y − y z + x z
dσ.
1 + 4x2 + 4y 2
0 %,4#55)
dσ =
1 + f 2x + f 2y ds =
1 + 4x2 + 4y 2 ds.
/ z = x2 + y2 0
=
$$
(−2x2 y 2 + (x2 + y 2 )(x2 − y 2 ))dσ,
σxy
σxy 6 R
#
= 2
Oxy &
=
$$
(−2r4 cos2 ϕ sin2 ϕ + r4 (cos2 ϕ − sin2 ϕ))rdrdϕ =
σxy
$2
sin2 2ϕ
=
+ cos 2ϕ dϕ r5 dr =
−
2
0
0
2π 6 2
2
ϕ sin 2ϕ sin 2ϕ r
16π
=
+
+
6 = − 3 .
4
16
2
0
$2π
0
F
ACBA
! P = 2x + z" Q = y + 2z" R = z − y
" " # $ % &'(
∂R ∂Q
−
i+
∂y
∂z
rot F =
∂P
∂R
−
∂z
∂x
j+
∂Q ∂P
−
∂x
∂y
k=
= (−1 − 2)i + (1 − 0)j + (0 − 0)k = −3i + j.
n
) *
$$
$$
n rot F dσ =
σ
5
2
(−1 − )dσ = −
3
3
ACB
=−
rot F ABC
5 SABC
5
=−
3 cos γ
3
$$
5
dσ = − dσACB =
3
ACB
1
OA
2
· OB
536·3
=−
= −22,5.
2/3
32 2
+ , " # $ % & ( -$-
$$ ACBA
&
$$
n rot F dσ = −22,5.
F dl =
ACBA
ACB
%
. -$-
F dl /
ACBA
P dx + Qdy + Rdz , AC 0 CB BA
• AC y = 0 #z = 3 − x2
dy = 0 dz = − 12 dx
(2x + y)dx + (y + 2z)dy + (z − y)dz =
#0 7
6
4
x+
3
2
dx =
7
8
AB
0
x2 + 32 x 6 = −40,5.
dx = 0 dz = dy
• CB x = 0 # z = 3 + y
(2x + y)dx + (y + 2z)dy + (z − y)dz =
−3
#
CB
(9 + 3y) dy = 9y +
0
3y 2
2
−3
= −13,5.
0
dz = 0 dx = 2dy
• BA z = 0 # x = 6 + 2y
(2x + y)dx + (y + 2z)dy + (z − y)dz =
BA
2 0
(24 + 9y) dy = 24y + 9y2 = 31,5.
−3
−3
%
F dl = −40,5 − 13,5 + 31,5 = −22,5
#0
ACBA
F = (2xy + z 2 )i + (x2 + z)j + (y + 2xz)k
!
" P = 2xy + z 2 # Q = x2 + z # R = y + 2xz $
% & '()
∂R ∂Q
∂P
∂R
−
= 1 − 1 = 0;
−
= 2z − 2z = 0;
∂y
∂z
∂z
∂x
∂Q ∂P
−
= 2x − 2x = 0,
∂x
∂y
% rot F = 0 *+
, &('-.) / x0 = y0 = z0 = 0 %
0
+ , P (x; y; z)# Q(x; y; z) R(x; y; z)
$x
$y
$z
2
u(x; y; z) = 0dx + x dy + (y + 2xz)dz = x2 y + yz + xz 2 + C.
0
0
0
∂u
∂u
∂u
= 2xy + z 2 # Q =
= x2 + z # R =
= y + 2xz.
$ P =
∂x
∂y
∂z
F = xi+yj+zk
x = −a x = a
y = −a y = a z = −a z = a
F = xyi + yzj + xzk
! " x2 + y2 + z2 = 1
#"
F = yzi + zxj + xyk $
!"#
$ %
&
' & %
( )
)
%
L
$" L = AB " % ' ' *
'
+ S
Z , -./
z
y
zn
B
zi
A
z0
z i-1 ζ
i
S
x
n !
" !
" " L# # zA = z0 zB = zn $!
% L " # &" '
( ") zi−1 zi * " ξi n
f (ξi )(zi − zi−1 ) = f (ξi )Δzi
" ) " "" f (ξi)Δzi
L
i=1
+, f (z)
n z L
f (ξi)Δzi
i=1
! "
L "
ξi #
$
f (z)dz = lim
n→+∞
L
n
+,
f (ξi )Δzi .
i=1
$ % & )
+, " ) L " !
# "
" # z
* -" f (z) ./
# z = x + iy# f (z) = u + iv# ) u = u(x; y)# v = v(x; y) 0 !
( z f (z) ) # " # i2 = −1#
$" $
$
$
f (z)dz = (u + iv)(dx + idy) = udx − vdy + i vdx + udy. +/
L
L
L
#
L
#
1 # Re f (z)dz = u(x; y)dx−v(x; y)dy
L
L
#
#
Im f (z)dz = v(x; y)dx + u(x; y)dy )
L
L
L ) !
2 3 ) # "
2 )
L
#
z = 1 + 2i
Imzdz
L
y
z
2
z
1
ϕ
Β
x
1
1
0
Α
1
y = 2x dy = 2dx
Imz = Im(x + iy) = y
$
$1
$
Imzdz =
0z
y(dx + idy) =
0z
$1
2xdx + i
0
4xdx =
0
1
1
= x2 0 + i · 2x2 0 = 1 + 2i.
# |z|dz
L
L
z = cos ϕ y = sin ϕ (0 ϕ π)
z = cos ϕ + i sin ϕ dz = (− sin ϕ + i cos ϕ)dϕ
|z| =
x2 + y 2 + cos2 ϕ + sin2 ϕ = 1.
$
$π
(− sin ϕ + i cos ϕ)dϕ = cos ϕ|π0 + i sin ϕ|π0 = −1 − 1 = −2.
|z|dz =
AB
0
!" #$%
#& '
&
dz
z
L
y
z
1
|z|=1
0
1
x
( ) * + z , z = |z|eiϕ = eiϕ
!" ''% dz = ieiϕ dϕ (0 ϕ 2π) -
&
dz
=
z
|z|=1
$2π
0
ieiϕ dϕ
= iϕ|2π
0 = 2πi.
eiϕ
#& .
! "# f (z) = u(x; y) + iv(x; y)% " $
S % f (z)dz & ' ! " !!
L
$
Re
#
L
zA zB
#
f (z)dz =
L
L
f (z)dz
LAB
# #
f (z)dz = vdx + udy
udx − vdy Im
L
L
! "#$ $ !
$
P dx + Qdy
LAB
% ! L %
#$ & #' " "#$ $%
∂P
=
() ∂Q
∂x
∂y
!
$
udx − vdy : P = u, Q = −v →
∂v ∂P
∂u
∂Q
=− ;
=
,
∂x
∂x ∂y
∂y
L
!
$
vdx + udy : P = v, Q = u →
∂u ∂P
∂v
∂Q
=
;
=
.
∂x
∂x ∂y
∂y
L
* & $ + f (z) = u+iv "
∂v ∂u
∂v
=
, $% & - . /0 ∂u
1 = − ∂x
∂x
∂y ∂y
() $%# $ 2 #
" ! f (z)dz %
L
! %
$ & 3
#
! f (z)dz # 4 "
L
&
f (z)dz = 0.
5
L
* % & !
" + ! $' # ' + 5 %
# % % n
#
n
n
f (z) 3x
S L0 L1 L2
!"!#
y
Z
L2
B
L1
B
B
H
A
B
C
B
H
H
D
H
L0
S
x
$% A ∈ L0 % B ∈ L1 C ∈ L1
% D ∈ L2 & % ' '
&
L = A Lo A B L1 C D L2 D C L1 B A ,
f (z) ( ) % * + *+
, $
&
$
$
$
$
$
f (z)dz =
+
+
+
+
+
L
A L0 A
A B
$
B L1 C
$
+
+
D C
C D
D L2 D
$
+
C L1 B
= 0.
B A
#
#
# # #
#
%
= −
= −
+
= − f (z)dz - &
A B
B A C#D
L1
% D C B L1 #C C L%1 B
= f (z)dz
= f (z)dz
D L2 D
L2
A L0 A
L0
&
&
&
f (z)dz =
L0
n
f (z)dz +
L1
L2
&
f (z)dz =
n &
k=1 L
L0
f (z)dz.
!
f (z)dz
k
"#"
$
f (z) S
L0
Lk
% &' "( )
* +
&
%
!" # (z − a)mdz L $
L
" m $
, - . % a L &(
'( (z − a)%m (( ( L " m
(z − a)m dz = 0
L
Z
y
|z−a|=δ
a
S
L
x
δ
L
a
a
L
m ≥ 0 %
(z − a)mdz = 0
L
m < −1
a
!
" L # # δ
a
# L $ %&'( ) δ
C : |z − a| = δ z = a + δeiϕ
S
* (z − a)m 2x +
L C
, -.'
&
&
m
m
(z − a) dz = (z − a) dz =
L
C
&
=
(z−a)=δ
dz
=
(z − a)k
$2π
0
m = −1
k = −m > 0
z − a = δeiϕ
dz = iδeiϕ dϕ
iδeiϕ dϕ
= iδ 1−k
δ k eikϕ
$2π
0
=
dϕ
=
ei(k−1)ϕ
=
2π
e−i(k−1)ϕ
iδ 1−k
(1 − k)i 0
= 0,
ez T = 2πi
m = −1 $-.&(
&
|z−a|=δ
/
dz
=
z−a
$2π
0
iδeiϕ dϕ
= iϕ|2π
0 = 2πi.
δeiϕ
&
(z − a) dz =
m
2πi, m = −1,
0, m = −1.
$-.-(
L
f (z)
a 0 +
L
! "
a
%
f (z)
1
2πi
f (z)dz
L
Resf (z) =
1
2πi
&
f (z)dz.
L
f (z) a
f (z) = · · · +
C−1
C−2
C−1
+
+ ··· +
+
(z − a)n
(z − a)2 z − a
+C0 + C1 (z − a) + · · · + Cn (z − a)n + . . .
!" #$ % & 0 < |z − a| < R' %
%' (
#% R
a' $ a !" L "
0 < |z−a| < R ! ) ( ( *
&
&
&
&
+∞
+∞
dz
dz
+
f (z)dz =
C−m
+
C
C
(z − a)k dz.
−1
k
(z − a)m
z − a k=0
m=2
L
L
L
L
+
& ) (
% dz ) # (
( # ' z−a
= 2πi' # ' ,
L
%
" ' f (z)dz = C−1 2πi' #
L
#
C−1
* $
$ & $ a *,
f (z)
-
Resf (a) = C−1 .
. &#% f (z)' ,
) $ & $ ,
/$ ) $ ' ( n,) ( 0
) " n ' / & $ $
0 & + ) ' '
1 a ( n,) ( f (z)'
* $ 2
C−n
C−n+1
C−1
+
+
+ ··· +
Ck (z − a)k .
(z − a)n (z − a)n−1
z − a k=0
+∞
f (z) =
(z − a)n
(z − a) f (z) = C−n + C−n+1 (z − a) + · · · + C−1 (z − a)n−1 +
n
+
+∞
Ck (z − a)k+n .
k=0
n − 1
n−1
d
((z − a)n f (z)) = C−1 (n − 1)! + C0 n(n − 1) . . .
dz n−1
. . . 2(z − a) + C1 (n + 1)n . . . 3(z − a)2 + . . .
z → a
dn−1
((z − a)n f (z)) = C−1 (n − 1)! = (n − 1)!Resf (a).
z→a dz n−1
f (z)
lim
n
Resf (a) =
dn−1
1
lim n−1 ((z − a)n f (z)) .
(n − 1)! z→a dz
!"#$%
5
!"#!
f (z) = (z +z 1)4
1
& ' ( ) f (z)
= (z+1)
z = −1
z
z
*+ f (z) = (z+1)
,
- *+ # - *.%# !"#$%
4
5
5
4
z 5
d3
1
lim
=
Res
(z + 1)4 z=−1 3! z→−1 dz 3
=
(z + 1)4
z5
(z + 1)4
=
1
1
5·4·3
lim (z 5 )III =
lim 5 · 4 · 3z 2 =
= 10.
z→−1
z→−1
3!
3!
1·2·3
/ a
0 0! = 1 f 0 (z) = f (z) !"#$%
n = 1
Resf (a) = lim (z − a)f (z).
!"#"%
z→a
!"#1
z2
= 2
z −1
f (z) =
1
f (z)
= z z−1 = (z−1)(z+1)
z
a1 = 1 a2 = −1 f (z) = z z−1
!!"
2
2
2
2
2
z2
1
z 2
z2
= lim
= ,
= lim(z − 1) 2
2
z − 1 z=1 z→1
z − 1 z→1 z + 1
2
z2
z 2
1
z2
Res 2
= lim
=− .
= lim (z + 1) 2
z→−1
z −1
z − 1 z→−1 z − 1
2
Res
z=−1
ϕ(z)
# f (z) f (z) = ψ(z)
$ ϕ(z) ψ(z) %
a & ϕ(a) = 0 a
& ' ψ(z) ψ(a) = 0 ψ (a) = 0
' !("
Resf (a) = lim (z − a)
z→a
ϕ(z)
= ϕ(a) lim
z→a
ψ(z)
) * '
ϕ(a) = 0 ψ(a) = 0 ψ (a) = 0
Res
1
ψ(z)−ψ(a)
z−a
a
ϕ(z)
ϕ(a)
.
=
ψ(z) z=a ψ (a)
!-
=
ϕ(a)
.
ψ (a)
f (z) =
ϕ(z)
ψ(z)
!+,"
f (z) = z z−1
a1 = 1
a2 = −1 . ϕ(z) = z2 * / .
!+,"
2
2
Res
z 2
z 2
z 2
1
z 2
1
;
Res
=
=
=
=− .
2
2
z − 1 z=1
2z z=1 2
z − 1 z=−1
2z z=−1
2
# a f (z)
f (z)
C−1
!( f (z) = e1/z
z = 0
) z = 0 / 0 * 0 0
f (z) = e1/z 1 &
234
e1/z = 1 +
1
1
1
+
+ ··· +
+...
z 2!z 2
n!z n
C−1 = 1
Res e1/z z=0 = 1
f (z)
S
L !
ak ∈ S k = 1, 2 . . . , n
L "# $ "
f (z) % # 2πi
&
f (z)dz = 2πi
n
Resf (ak ).
k=1
L
ak
! γk " ! # $ %$
& " ! # #
y
z
a2
a1
an
γ2
γ1
γn
S
L
x
' #! ( # ( L γ1 γ2 . . . γn
) * ! f (z) ( & +
&
f (z)dz =
& ,
L
n &
k=1 γ
f (z)dz,
k
&
f (z)dz = 2πiResf (ak ),
γk
&
z4
|z|=3/2
ez dz
+ 3z 2 − 4
z
e dz
! f (z) = z4 +3z
"
2 −4
4
2
$ z + 3z − 4 = 0 % z 4 + 3z 2 − 4
% z 4 + 3z 2 − 4 = (z − 1)(z + 1)(z − 2i)(z + 2i) $
# a1 = 1 a2 = −1 a3 = 2i a4 = −2i !!&! '
' & ( % |z| = 3/2 $ !!
a1 a2 a3 a4 % ) *
#
= 0
"
$
y
Z
2
|z|=1
|z|=3/2
1
-1
1
0
2 x
-2
+
"
&
|z|=3/2
ez
+
Res 4
z + 3z 2 − 4 z=1
ez
.
+ Res 4
z + 3z 2 − 4 z=−1
ez dz
= 2πi
z 4 + 3z 2 − 4
z = 1
! !
Res
z4
ez
ea
,
= 3
2
+ 3z − 4 z=a 4a + 6a
z = −1
e
Res 4
z + 3z 2 − 4
a=1
z
z=1
e
;
10
a = −1
e
Res 4
z + 3z 2 − 4
z
=−
z=−1
&
|z|=3/2
ez dz
= 2πi
4
z + 3z 2 − 4
e
1
−
10 10e
=
e−1
1
=−
.
10
10e
πi 2
(e − 1).
5e
!" #$
$2π
0
dϕ
(a > 1)
a + cos ϕ
% & ' (
e = z dz = ieiϕ dϕ = izdϕ cos ϕ =
z 2 +1
= 2z )
ϕ
0 2π z
=
|z| = 1 *
iϕ
eiϕ +e−iϕ
2
$2π
0
dϕ
=
a + cos ϕ
=
&
|z|=1
dz
2
=
z 2 +1
i
iz(a + 2z )
&
dz
=
z 2 + 2az + 1
|z|=1
z 2 + 2az + 1√= 0,
1
=
z1,2 = −a ± a2 − 1 , Res z2 +2az+1
z1
z1 ∈ |z| < 1,
z2 ∈ |z| > 1,
= 2√a12 −1
1
2z+2a z1
=
2
1
2π
= 2πi · √
=√
.
i 2 a2 − 1
a2 − 1
+ & ,
-
,
N .
#/!0 N → +∞
y
z
N
aj
a2
a1
N
x1
an-1
an
0 xk xm-1 x m N x
x2
f (z)
aj
f (z)
Imz ≥ 0
j = 1, 2, . . . n Imaj > 0
$+∞
n
f (x)dx = 2πi
Resf (aj ).
−∞
!
j=1
$+∞
−∞
x2 + 1
dx"
x4 + 1
!" f (z) = zz +1
#! $ %
+1
& '( ! ) * !
+ ,,- !" ,,
2
4
√
a1,2 =
a21,2 + 1
z 2 + 1
2
(i ± 1); Res 4
=
=
2
z + 1 a1,2
4a31,2
1 2
(i ± 2i + 1) + 1
2
4
√ (i3 ± 3i2 + 3i ± 1)
2 2
√
√
√
21±i i±1
2(i ± i2 ± 1 + i)
2i
±i + 1
= 4
·
=
=−
.
=
2
√ (i ∓ 1)
4 i∓1 i±1
4(i − 1)
4
2
=
$+∞
−∞
√
√
√
√
x2 + 1
2
2
2
dx = 2πi −
i−
i = 2πi −
i = π 2.
4
x +1
4
4
2
!"
# $ % & '#% (
f (z) = g(z)eiαz α > 0 g(z) → 0
|z| → +∞
Imz ≥ 0 aj j = 1, 2, . . . , n (Imaj > 0)
xk k = 1, . . . , m x !"#
n
$+∞
m
1
f (x)dx = 2πi
Resf (aj ) +
Resf (xk ) .
2 k=1
j=1
)
−∞
$+∞
$
−∞
sin x
dx
x
$+∞
−∞
cos x
dx
x
* + , - . #/ (
' 0) 1 & - )))
ix
e = cos x + i sin x
$+∞
−∞
eix
dx =
x
$+∞
−∞
cos x
dx + i
x
$+∞
−∞
sin x
dx.
x
2 '#
# #
→ 0 |z| → +∞ 4
# 3 α =
e
5 z = 0 Res z z=0 = e1 z=0 = e0 = 1
iz
f (z) = ez
1 g(z) = 1z
iz
$+∞
−∞
cos x
dx + i
x
$+∞
−∞
sin x
1
eiz
dx = 2πi Res
= πi.
x
2
z z=0
4 #
$+∞
−∞
iz
5 5
cos x
dx = 0,
x
6
$+∞
−∞
sin x
dx = π.
x
$+∞
−∞
sin x
dx
x4 − 1
$+∞
−∞
cos x
dx.
x4 − 1
$+∞ ix
$+∞
$+∞
e
cos x
sin x
dx =
dx + i
dx.
x4 − 1
x4 − 1
x4 − 1
−∞
−∞
−∞
iz
f (z) = ze4 −1 !"
# α = 1" g(z) = z41−1 → 0 z → +∞ " z 4 − 1 =
= (z − 1)(z + 1)(z 2 + 1) $ %
x : a1 = 1"
a2 = −1 &# ' % a3 = i $ '
( $ $ )* + ,!
cos 1 + i sin 1
eiaj
ei
eiz
;
Res 4
= 3 ; Resf (1) = =
z −1
4a
4
4
z=aj
j
− cos 1 + i sin 1
e−1
i
e−i
=
; Resf (i) =
= .
Resf (−1) = −
4
4
−4i
4e
- + ,!
$+∞
$+∞
cos x
sin x
dx + i
dx =
x4 − 1
x4 − 1
−∞
−∞
1
= 2πi Resf (i) + (Resf (1) + Resf (−1)) =
2
i
cos 1 + i sin 1 − cos 1 + i sin 1
π
= 2πi
+
= − − 2π sin 1.
4e
2
2e
.
$+∞
$+∞
1
cos x
sin x
dx = −π
+ 2 sin 1 ,
dx = 0.
x4 − 1
2e
x4 − 1
−∞
−∞
$1
$3−x
dx
0
√
2y(x+2y 2 )dy
1−x2
z = x2 x = y2 x = 1 z = 0
&
(3xy + x2 )dx + 8x2 dy
ABCA
A(0; 1)! B(2; 2)! C(0; 3)
" #
$" %
!
F = (x + y; x + y + z; 2z − y) & #
O(0; 0; 0)! A(−3; 0; 0)! B(0; 2; 0)! C(0; 0; 3)
$" ' %
(
F = (ex sin y; ex cos y; 1)
" # $
$1
$3−x
dx
√
1−x2
0
$1
=
0
2
$1
2y(x + 2y )dy =
3−x
(y 2 x + y 4 )√1−x2 dx =
0
(3 − x)2 x + (3 − x)4 − x(1 − x2 ) − (1 − x2 )2 dx =
$1
=
(9 − 6x + x2 )(9 − 5x + x2 ) − x + x3 − 1 + 2x2 − x4 dx =
0
$1
=
(81 − 54x + 9x2 − 45x + 30x2 − 5x3 + 9x2 − 6x3 + x4 − x + x3 − 1+
0
2
4
$1
+2x − x )dx =
(80 − 100x + 50x2 − 10x3 )dx =
0
=
80x − 50x2 +
1
265
50 4 10 4
50 5
x − x = 30 +
− =
.
3
4
3
2
6
0
x = 0 x = 1 y = 3 − x
y 2 = 1 − x2 ! ABCE
"#$
y
3
2
1
C
x=3-y
K
E
B
D
A
0
1
x= 1-y 2
2
3
x
ADE EDKB KBC
ABCE
$1
$3−x
dx
√
0
$2
+
2y(x + 2y )dy =
$1
2y(x + 2y 2 )dx +
0
$3
=
+
0
$1
$1
2y(x + 2y 2 )dx =
dy
√
0
1−y 2
2
3−y
yx + 4y 3 x 0 dy +
$2
2
$1
$3−y
dy
2y(x + 2y 2 )dx+
2
1−x2
dy
1
$3
2
1
yx2 + 4y 3 x 0 dy+
1
1
yx2 + 4y 3 x √1−y2 =
$3
y(9 − 6y + y 2 ) + 4y 3 (3 − y) dy+
2
0
$2
+
(y + 4y 3 )dy +
1
$1
(y + 4y 3 − y(1 − y 2 ) − 4y 3
1 − y 2 )dy =
0
$3
2
3
4
$2
(9y − 6y + 13y − 4y )dy +
=
2
(y + 4y 3 )dy+
1
3
9y 2
13y 4 4y 5
− 2y 3 +
−
+
2
4
5 2
1
2
5 4 4 (1 − y 2 )5 4 (1 − y 2 )3
y2
4
+y +
y −
+
=
2
4
5
3
1
$1
5y 3 − 4y 3 1 − y 2 dy =
+
0
+
0
1053 972
81
− 54 +
−
− 18 + 16 − 52+
=
2
4
5
128
1
5 4 4
265
+
+ 2 + 16 − − 1 + + − =
.
5
2
4 5 3
6
OABCD
OCD z = x2 OAD OBC
x = y 2 x = 1 z = 0
!"# $%
z
y
b)
D(1;1;−1)
x=Const
a)
C(1;1;1)
y= x
B
OX
1
y
A(1;−1;0)
0
x
S A
B(1;1;0)
y=− x
x
& ' ( ) *+
'
!"# .%
$$
$$
V =
zds =
S
x2 dxdy = 2
$1
, (
√
$x
$1
√x
x2 dx dy = 2 x2 y 0 dx =
0
S
$1
=2
0
0
0
1
√
2
4
x2 xdx = 2 · x7/2 = .
7
7
0
/ y = 0 0 ) , x 1 2
3!"%
x =
7
4
$$$
xdv =
V
7
4
-
√
$1
$x
xdx
0
$x2
dy
dz =
√
− x
0
0
− x
7
4
$1
0
3!%
x2
√
$ x
xdx
z dy =
√
− x
0
√ x
$1
$x
$1
$1
√
7
7
7
=
x3 dx
dy =
x3 y
dx =
2x3 xdx =
4
4
4
√
√
√
0
− x
0
z
1
7 2 9
7
· x2 = ;
2 9
9
0
=
z =
7
4
$$$
7
4
zdv =
V
=
7
8
$1
√
$1
$x
dx
√
x4 dx
$x
dy =
√
− x
0
=
$$
=
$x2
dy
√
− x
0
7
8
$1
0
zdz =
0
7
4
$x
dx
√
− x
0
√x
7
x4 y −√x dx =
4
$1
x 2
z 2
dy =
2 0
√
x4 xdx =
0
1
7
7 2 11
x2 = .
4 11
22
0
&
∂Q ∂P
−
∂x
∂y
√
$1
P dx + Qdy =
L
dxdy
y
3
C
x=Const
S
z
y=3-
2 S
B
1A
y=1+
2
3
1
x
2
C
n
A
-3
1
x
2
B
2
0
x
y
x
P = 3xy + x2
&
2
Q = 8x2
∂Q
∂x
−
ABCA
13xdxdy = 13
0
3− 12 x
$
xdx
0
S
= 13
= 16x − 3x = 13x
$2
$$
2
(3xy + x )dx + 8x dy =
$2
∂P
∂y
dy =
1+ 12 x
$2
x
x
dx = 13 (2x − x2 )dx =
x 3− −1−
2
2
0
2
52
x
8
= .
x2 −
=
13
4
−
3 0
3
3
3
= 13
AB : y = 1 + 12 x dy = 12 dx
BC : y = 3 −
dy = − 12 dx CA : x = 0 dx = 0
!" #$
1
x
2
&
$2
x
3x 1 +
+ x2 + 4x2 dx+
(3xy + x )dx + 8x dy =
2
2
2
0
ABCA
$0
+
2
$0
$2
x
3x 3 −
+ x2 − 4x2 dx + 0dy = (−6x + 11x2 )dx =
2
3
=
0
2
11
52
−3x2 + x3 = .
3
3
0
%& '
( ) *#+, - $#. &
V = 61 OA · OB · OC =
1
6
·2·3·3=3
/ * 01 0
F $ - σ
2σ
$$
$$$
div F = ∂(x+y) + ∂(x+y+z) +
∂x
∂y
= F ndσ =
divF dV = ∂(2z−y)
∂z = 1 + 1 + 2 = 4
σ
V
$$$
dV = 4V = 12.
=4
V
=
z
x=const
3 C
z=x+3
A
-3
0 x
OAC
n=
N
N
= − √217 ; √317 ; √217
3
2
1
2
F n = − √ (x + y) + √ (x + y + z) + √ (2z − y) = √ (x − y + 7z) =
17
17
17
17
ABC
=
=
z = 3 + x − 32 y
23
21
1
1
=√
x − y + 21 + 7x − y = √ (8x − y + 21),
2
2
17
17
$$
1
23
√
8x − y + 21 dσ =
2
17
ABC
= dS = dxdy = dσ · cos γ =
$2
$0
21
23
4x − y +
dxdy = dy
4
2
$$
=
0
OBA
$2
=
2x2 −
−2
=
0
$2
−
=
0
3
y−3
2
=
23
21
4x − y +
dx =
4
2
3
y−3
2
0
21
23
y + x
dy =
4
2
3
y−3
2
0
$2
√2 dσ
17
2
23
+ y
4
3
21
y−3 −
2
2
3
y−3
2
dy =
69y 2 69
63
63
−9y 2
+ 18y − 18 +
− y− y+
dy =
2
8
4
4
2
$2
=
0
33 2
27
y − 15y +
dy =
8
2
2
11 3 15 2 27
y − y + y =
8
2
2
0
= 11 − 30 + 27 = 8,
$$
F ndσ = 2 + 2 + 0 + 8 = 12.
σ
F = F (x; y; z)
rot F = 0 !" #
$
k
i
j
∂1 ∂ex cos y
∂
∂
∂
=
−
i+
rot F = ∂x
∂y
∂z
∂y
∂z
ex sin y ex cos y 1
∂ex cos y ∂ex sin y
∂1 ∂ex sin y
+
j+
−
k=
+ −
∂x
∂z
∂x
∂y
= 0 · i + 0 · j + (ex cos y − ex cos y)k = 0.
% & ' $ ()" & * +
, , x0 = y0 = z0 = 0 $
$x
$y
$z
x
V (x; y; z) = 0dξ + e cos ηdη + dζ + C =
0
x
sin η|y0
x
0
ζ|z0
0
x
+
+ C = e sin y + z + C.
=e
∂V
∂(e sin y + z + C)
=
= ex sin y = Fx
% & *
∂x
∂x
∂V
∂(ex sin y + z + C)
=
= ex cos y = Fy ;
∂y
∂y
∂V
∂(ex sin y + z + C)
=
= 1 = Fz .
∂z
∂z
$1
1−x
$ 2
dx
−1
√
− 1−x2
xydy
x2 + y2 = z + 1 z = 3
% 2xydx + (x2 − 3xy)dy A(1; 1)
ABCA
B(2; 2) C(3; 1) ! " ABC
#$ %
F = (2z; x−y+z; 3y+2z)
& " O(0; 0; 0) A(−4/3; 0; 0)
B(0; 2; 0) C(0; 0; 4) #$ '
%
(
F = (2xz + y 2 ; 2xy + z 2 ; 2yz + x2 )
Oxy
x + y = 4 Oxy
x + y = 3 (x + 1) +
+(y − 1) = 4 (x + 3) + y = 1
+ y = 1 z x + 9y + 18y + 9
z < x + 2x + y /4 − y/2 + 5/5
2x − z − 1 > 0 y + 2x + 1 > 0
(x + 3) + y = c
+
=1
c > 18 ! "
(x − 1) + (y − 1) = 4
! " + y = 1 # ! "
x − 1 = (y + 1) #
! " y − 1 = x
+ z − = 0.
! # Oz.
$ ! # Oy. !
# Oy. ! # %
Oz. & #
! # Ox.
(x + 1) + +
= 1 '( (x − 1) +
+(y + 1) + z = 3 ! # − + z = 1
! # (y − 1) − + z = 1 $
! # +z − = −1 $ !
= −1
# (x − 1) + y −
! #
+
+
y =
! # x =
! # z = − − (y + 1) &
# ! # y = − $ !
# + Y − = −1 λ = 3 λ = 6 λ = −2 det A = −36
det D = 216
+ + = 1 λ = 2 λ = 5
λ = 8 det A = 80 det D = 2560
+ + = 1
λ = 1 λ = 2 λ = 3 det A = 6 det D = −42.
2
2
2
2
2
2
2
2
x2
4
2
2
2
2
2
2
x2
4
2
y2
4
2
(z−1)2
1/2
y2
1/4
2
x2
2
x2
3
2
2
x2
3
(z+1)2
3
2
z2
3
x2
2
X2
2
y2
2
2
2
∂z
∂x
4
2
x2
2
Z2
3
2
9
z2
3
1
X2
16
2
Y2
32/5
y2
2
(y+1)2
4
2
x2
3
Z2
4
3
1
(y+1)2
c−18
2
2
2
2
x2
9(c−18)
2
2
x2
4
2
1
X2
7
2
(z−1)2
4
2
Y2
7/2
Z2
7/3
3
= 4x − 2y + 3;
∂z
∂y
= −6y − 2x − 5.
∂z
∂x
=
−3(2x−5y)−2(2y−3x)
;
(2x−5y)2
ln (
√
x+1)12
x
12
−
12
√
x
12
+ C. −
√
(3−4x)(9−5x)
5
+ 2021√5 ln |51 − 40x +
+ C. √16 ln |12x +
5(3 − 4x)(9 − 5x)| + C. − 3 x+1
x−2
√
2
√
7 + 8x − 11x2 + 1163
·
+5 + 2 6(6x2 + 5x + 11)| + C. − 11
11
+4
√
1 15+3x
√
+ C. arccos x√1 2 + C. − 15
+ C.
· arcsin 11x−4
x
93
3
x
1
3+4x
−3
− 5√x2 −5 + C. x
. 3x + 4 = t2 .
x
h
n−1
f (a + ih), h =
i=0
1
2
(a
2
b−a
.
n
heh
eh −1
2
eb − ea , h =
− b2 ) . π6 . e3 − e2 .
2
1 + ln 94 . π4 . πa4 .
−2.
ln
π
2
− 1.
b−a
.
n
√
1+ 5
.
2
π
.
2
2
323 .√ 38π
. 32π
. 16 . πc2 .
R2 √
a2
√
x
2
2
2π 5. 2 4a x + 1 + 4a1 ln
2ax + 4a2
x2 + 1 .
6R. a2 ϕ + 1 + ϕ2 + ln ϕ + 1 + ϕ2 .
π
.
3
gt2
.
2
√
2abπ . 2 2.
π8 . π.
2
arctg 2√tg3x + C xex + C 29
arctg 5x+3
−
45
9
1+x 1
1
ln(5x + 6x + 18) + C − 4 ln 1−x − 2 arctg x + C
3
tg3 x + C 187 6π
12 γπHR4
3
− 10
1
√
2 3
2
√
ln(25/24). 76 y =
2x − x2 ,
2
y = 4x, x = 0, x = 2. x = y /2, x = 3 − y 2 , y = 0, y = 1.
√
1− 1−y 2
√
√
#1
#
2
y = 2x − x , y = 4x, x = 0, x = 2, dy
f (x, y) dx +
√
14π/3.
+
#1
0
dy
1+
#
√
1−y 2
f (x, y) dx +
√
2# 2
1
0
dy
#2
y 2 /4
f (x, y) dx.
y 2 /4
x
=
y 2 /2,
√
1/2
#2x
#
# 2 #1
3 − y 2 , y = 0, y = 1, dx f (x, y) dy +
dx f (x, y) dy +
x =
√
√
+
#3
√
dx
0
3−x
# 2
0
1/2
0
f (x, y) dy.
0
2
π/4
#
dϕ
1/ #
cos ϕ
0
0
3π/4
#
ρf (ρ cos ϕ, ρ sin ϕ) dρ+
dϕ
dϕ
1/#
sin ϕ
0
ρf (ρ cos ϕ, ρ sin ϕ) dρ.
0
π/4
1/#
sin ϕ
π/4
π/2
#
ρf (ρ cos ϕ, ρ sin ϕ) dρ.
4 4 3
π a.
3
2
π(1−e−a ).
√
3(π/4 + 1/2. 17/24.
x = 5a/6, y = 0. 21
πa4 γ, 49
πa4 γ,
32
32
dx
1−x
#
√
# 1−x−y
R#2 −x2
#R
#H
f (x, y, z)dz.
dx
dy f (x, y, z)dz.
0
√
0
0
0
−R
− R2 −x2
√
√
√
√
5
8
15 (31 + 12 2 − 27 3. 4π 2/3. πa5 (18 3 − 976 ).
83 R3 (π − 43 ). π/10.
#1
35
πa4 γ.
16
dy
z = 4/3, x = y = 0, x = y = 0, z = 2H/3.
πH 2 (R24 − R14 )/4. 14 πH 2 (R24 − 3R14 + 2R12 R22 ), 14 πH 2 (3R24 −
−R14 − 2R12 R22 ).
40 19
−2πa2 − 43
30
ex−y (x + y) + C ln |x + y| + C
24a3
#1
−1
dx
1−x
# 2
√
− 1−x2
3π
16
3
πa2
8
x3 −x2 y−y 3 +C
u = xyz + C
√
√
2
2
#1−y
#1
#1−y
xydy =
dy
xydx + dy
xydx = 0
√
√
−1
0
x = y = 0 z =
= xy 2 + yz 2 + x2 z + C
#0
−
5
3
1−y 2
−4
−
16
9
1−y 2
U (x; y; z) =
," "
$"
," " $"
#
," " "
##
," " "
#
-("
%
- . ( "
( * .
- .
%#
- ""
( *
#
- "
$"
- "
$" #
- )
.
-
"
&
-
/ . . %%
/ %
/ *
/
&
!"
#
!" "
$ #%
!
! &
! &
! %
' &%
'
' " &
' " (
&
'
' )
&%
'
' *
&
'
' +""
#
!
"#
$ %& & ' (
% !"
$'
) *
$% )
!!
$+ "
$
#
$% ,$ -.%/+ !.%/+ 0) ,,,
.%/+ '
( () ,,!
. % ,
.+ #
1 %( *
1 %
1
1% ' ) ,
10
10 0) ,,
1 ) ,"
1 0)
1
1 0 *
1 *"
1) % 1 -,
1 #
1
#!
1 )
,2 ) (
"
1' %
,""
1' )
,!,
1' ) 3 -
4&
$3 #
4 )
5 ,
5 ) -
5 "
5 (
) ",
5 (
5( )
6 %
6 7 *,
6 . 27
6 5 -
6 0 )
,!
6 ,
8 ) ( )
*
8 !*
9'( +
,!
9 $3 2 % ,!"
9 5 ,#,
9 ,#,
:( )
:( "
:) )
; ' 2 )
##
; ' &
) #
Министерства образования и науки РФ «Лучшее учебное издание по математике
в номинации «Математика в технических вузах»
В. А. ЛЯХОВСКИЙ, А. И. МАРТЫНЕНКО, В. Б. МИНОСЦЕВ
КУРС МАТЕМАТИКИ
ДЛЯ ТЕХНИЧЕСКИХ
ВЫСШИХ УЧЕБНЫХ
ЗАВЕДЕНИЙ
Часть 2
Функции нескольких переменных.
Интегральное исчисление.
Теория поля
Под редакцией
В. Б. Миносцева, Е. А. Пушкаря
Издание второе, исправленное
ДОПУЩЕНО
НМС по математике Министерства образования и науки РФ
в качестве учебного пособия для студентов вузов, обучающихся
по инженерно&техническим специальностям
•САНКТ-ПЕТЕРБУРГ•МОСКВА•КРАСНОДАР•
•2013•
ББК 22.1я73
К 93
Ляховский В. А., Мартыненко А. И., Миносцев В. Б.
К 93
Курс математики для технических высших учебных
заведений. Часть 2. Функции нескольких переменных.
Интегральное исчисление. Теория поля: Учебное пособие /
Под ред. В. Б. Миносцева, Е. А. Пушкаря. — 2-е изд.,
испр. — СПб.: Издательство «Лань», 2013. — 432 с.:
ил. — (Учебники для вузов. Специальная литература).
ISBN 9785811415595
Учебное пособие соответствует Государственному образовательному
стандарту. Пособие включает в себя лекции и практические занятия.
Вторая часть пособия содержит 25 лекций и 25 практических занятий
по следующим разделам: «Дифференциальное исчисление функций
нескольких переменных», «Интегральное исчисление функций одной
переменной», «Кратные интегралы», «Криволинейные интегралы и
теория поля».
Пособие предназначено для студентов технических, физикоматематических и экономических направлений.
ББК 22.1я73
Рецензенты:
À. Â. ÑÅÒÓÕÀ äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðîôåññîð,
÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ;
À. À. ÏÓÍÒÓÑ ïðîôåññîð ôàêóëüòåòà ïðèêëàäíîé ìàòåìàòèêè è
ôèçèêè ÌÀÈ, ÷ëåí ÍÌÑ ïî ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è
íàóêè ÐÔ; À. Â. ÍÀÓÌΠäîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê,
äîöåíò êàôåäðû òåîðèè âåðîÿòíîñòåé ÌÀÈ; À. Á. ÁÓÄÀÊ äîöåíò,
çàì. ïðåäñåäàòåëÿ îòäåëåíèÿ ó÷åáíèêîâ è ó÷åáíûõ ïîñîáèé ÍÌÑ ïî
ìàòåìàòèêå Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ; Ó. Ã. ÏÈÐÓÌÎÂ
ïðîôåññîð, çàâ. êàôåäðîé âû÷èñëèòåëüíîé ìàòåìàòèêè è ïðîãðàììèðîâàíèÿ ÌÀÈ (Òåõíè÷åñêèé óíèâåðñèòåò), ÷ëåí-êîððåñïîíäåíò ÐÀÍ,
çàñëóæåííûé äåÿòåëü íàóêè ÐÔ.
Обложка
Е. А. ВЛАСОВА
Охраняется законом РФ об авторском праве.
Воспроизведение всей книги или любой ее части запрещается без письменного
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Любые попытки нарушения закона
будут преследоваться в судебном порядке.
© Издательство «Лань», 2013
© Коллектив авторов, 2013
© Издательство «Лань»,
художественное оформление, 2013
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h
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x
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−
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a2
b2
c2
c2
b2
! " #
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h2
R = b
− 1
c2
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( ) *
+
x2
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+
−
= 1,
5/4 5/3 5/2
Oy
9x2 − y 2 − z 2 = 5
! " # $
y2 z2
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+
−
= −1,
5
5
5/9
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&
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z=
x2 y 2
+ 2,
a2
b
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# Oxz Oyz #
*
x2
,
a2
y = 0,
z=
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x = 0,
z=
z = h (h > 0) +
√
√
x2
y2
+
= 1,
a2 h b2 h
z = h,
# a h b h ' &,) - a2 = b2
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Oyz
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z=
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2
a
b
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h
0
x
y
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a2 z = x2 ,
y = 0.
x = h
⎧
h2 y 2
z = 2 − 2,
a
b
x = h,
h2
b z− 2
2a
⎩
x = h.
⎨
2
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% $ '
z = h h = 0
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b
z = h,
h=
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−
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z = h.
* + ! % $ " '
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z
y
0
x
Oxy
Oxy
x2
y2
−
= 0
a2 ⎧
b2
x
⎨ + y = 0,
a b
⎩ x − y = 0.
a b
Oxy
x y
+ = 0,
a b
z = 0,
x y
− = 0,
a b
z = 0,
Oxy
! "
#
⎧ x
⎨ +
a
⎩ x−
a
y
= kz,
b
y
1
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b
k
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a
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a
y
1
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by
l
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b
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x2
z2
+
,
y=−
1/3 1/5
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' ' ( Oy
% # () ') y
1
1
' ( '
3
5
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x = 4z 2 − 16y 2 #
2
y2
z
−
,
1/4 1/16
+"
$, # Ox " $ # Oz " $
x=
$
2x2 + 3y 2 + 4z 2 − 5 = 0.
2x2 3y 2 4z 2
x2
y2
z2
+
+
=1⇔
+
+
= 1.
5
5
5
5/2 5/3 5/4
b=
5
c=
3
a =
5
2
5
4
3x2 − 4y 2 + 5z 2 − 6 = 0.
! "
#
3x2 2y 2 5z 2
x2
y2
z2
−
+
=1⇔
−
+
= 1.
6
3
6
6/3 3/2 6/5
#
# $
6
% # " Oy 3 65
y = 0
# " Oy
4y 2 + 4z 2 − 5x2 − 7 = 0.
! "
# &
4y 2 4z 2 5x2
y2
z2
x2
+
−
=1⇔
+
−
= 1.
7
7
7
7/4 7/4 7/5
# # ' $
% # " Ox 74 %
Ox
# ' " Ox
4y 2 + 5z 2 = 6x2 − 2.
6x2
z2
x2
5z 2
y2
− 3x2 = −1 ⇔
+
−
= −1.
2
1/2 2/5 1/3
2y 2 +
Ox
! " #$ Ox
%
3x2 + 3z 2 = 4y 2 − 4.
4y 2
3x2 3z 2
z2
x2
+
− y 2 = −1 ⇔
+
− y 2 = −1.
4
4
4/3 4/3
&
Oy
! " #$ Oy
'
3z 2 + 2y 2 − 5x = 0.
5x
%
x=
y2
z2
2y 2 3z 2
+
⇔x=
+
.
5
5
5/2 5/3
( )
Ox
! ) $ Ox
*
4y 2 − 3z 2 − 3x = 0.
Z
z
O1
Y
O
y
X
x
O1 XY
O1
x0 y0 z0 O1 (x0 ; y0 ; z0 )
!
" # $ %
& ' (%
x = X + x0 ;
y = Y + y0 ;
z = Z + z0 ,
)*"$
Y = y − y0 ;
Z = z − z0 .
)*!$
X = x − x0 ;
% +
& (& O Oxyz $
OXY Z $
,
' & x y z %
M % - X Y Z
. (
&
&
OX
% &
(% &
(%
cos ∠XOx = α11 ,
cos ∠XOy = α21 ,
cos ∠XOz = α31 .
L
+1
! " # $
! " #
L −1
! %&# ' (
Ox Oy ()
(
y
y
O
x
x
O
! !
" #$ % & '
! xy xz yz " a12 = a13 = a14 = 0$
( )
* %
* " +
3x2 + 2y 2 + z 2 − 6x + 4y − 4z + 5 = 0-
,
+ , - .&
(
-
3x2 + 2y 2 + z 2 − 6x + 4y − 4z + 5 = 0 ⇔
⇔ 3(x2 − 2x + 1) + 2(y 2 + 2y + 1) + (z 2 − 4z + 4) = 4 ⇔
⇔ 3(x−1)2 +2(y +1)2 +(z −2)2 = 4 ⇔
X = x − 1,
(x − 1)2 (y + 1)2 (z − 2)2
+
+
= 1.
4/3
4/2
4
Y = y + 1,
Z = z − 2.
2
X
Y 2 Z2
+
+
= 1.
4/3
2
4
√
2
a = √ , b = 2, c = 2.
3
X Y Z
x y z X = x − x0 Y = y − y0 Z = z − z0
! "#$!%!#& '
( '
P0 (x0; y0; z0)!
)
(
) P0 (1; −1; 2)
!
#*!+
4x2 + y 2 − 3z 2 + 16x + 2y + 6z + 6 = 0?
, -
'
(
4x2 + y 2 − 3z 2 + 16x + 2y + 6z + 6 = 0 ⇔
⇔ 4(x2 + 4x + 4) + (y 2 + 2y + 1) − 3(z 2 − 2z + 1) = 8 ⇔
⇔
(x + 2)2 (y + 1)2 (z − 1)2
+
−
= 1.
2
8
8/3
X = x + 2,
Y = y + 1,
Z = z − 1.
X2 Y 2
Z2
+
−
= 1.
2
8
8/3
OZ OZ
! "
# $ P0(−2; −1; 1)
Oz %
# !
& % '() *+ ,
!-- . a12 a13 a23
$
. $ - '() *+
! # . (, * /
0 % $ $
$ # 1
$
⎧
⎨ (a11 − λi )α1i + a12 α2i + a13 α3i = 0,
a12 α1i + (a22 − λi )α2i + a23 α3i = 0,
⎩
a13 α1i + a23 α2i + (a33 − λi )α3i = 0,
'(2 3+
'(2 (+
2
2
2
α1i
+ α2i
+ α3i
= 1,
$ # λ1 λ2 λ3 #
'(2 4+
a11 − λ
a12
a13
a12
a
−
λ
a23
22
a13
a23
a33 − λ
= 0.
'(2 4+
(2 5
⎛
⎞
a11 a12 a13
⎝ a12 a22 a23 ⎠ ,
a13 a23 a33
'(2 *6+
λi !"#$#%&' (
%)'*& # %)'+&
(X, Y, Z)
! " # $
%
a14 α11 +a24 α21 +a34 α31 =
$
= 0
& !
' ( & ! A $
) ( (det A = 0)
*
+
λ1 X 2 + λ2 Y 2 + λ3 Z 2 +
* + λ1 λ2 λ3 ,
&
⎛
det D
= 0,
det A
! )
⎞
a11 a12 a13
A = ⎝ a12 a22 a23 ⎠ ,
a13 a23 a33
det A , -
⎛
a11
⎜ a12
D=⎜
⎝ a13
a14
a12
a22
a23
a24
a13
a23
a33
a34
⎞
a14
a24 ⎟
⎟,
a34 ⎠
a44
!
det D , - & $
. &-
6x2 − 2y 2 + 6z 2 + 4zx + 8x − 4y − 8z + 1 = 0
⎛
6 0
⎝ 0 −2
2 0
⎞
2
0 ⎠
6
=0⇔
⇔ −(6 − λ)(2 + λ)(6 − λ) + 4(2 + λ) = 0 ⇔ (2 + λ) 4 − (6 − λ)2 = 0 ⇔
6−λ
0
2
0
−2
−
λ
0
2
0
6−λ
⇔ (2 + λ)(4 − 36 + 12λ − λ2 ) = 0 ⇔ (λ + 2)(λ − 8)(λ − 4) = 0 ⇔
⇔ λ1 = −2.
λ2 = 8,
λ3 = 4.
"
!
⎧
⎧
1
⎪
+
2α
=
0,
(6
−
4)α
α = −√ ,
⎪
⎪
11
31
⎪
⎪
⎨
⎨ 11
2
(−2 − 4)α21 = 0,
α21 = 0,
⇒
2α11 + (6 − 4)α31 = 0,
⎪
⎪
⎪
⎪
1
⎩ α2 + α2 + α2 = 1
⎪
⎩ α31 = √
11
21
31
2
⎧
⎧
1
⎪
(6 − 8)α12 + 2α32 = 0,
α =√ ,
⎪
⎪
⎪
⎪
⎨ 12
⎨
2
(−2 − 8)α22 = 0,
α22 = 0,
⇒
2α12 + (6 − 8)α32 = 0,
⎪
⎪
⎪
⎪
1
⎪
⎩ α2 + α2 + α2 = 1
⎩ α32 = √
12
22
32
2
⎧
⎧
(6
+
2)α
+
2α
=
0,
⎪
13
33
⎪
⎨
⎨ α13 = 0,
0 = 0,
α23 = 1,
⇒
2α13 + (6 + 2)α33 = 0,
⎪
⎩
⎪
α33 = 0.
⎩ α2 + α2 + α2 = 1
13
23
33
# L $ %
⎛
⎞
1
1
√ 0
−√
⎜
⎟
2
2
⎜
⎟
0 1 ⎟.
L=⎜ 0
⎝ 1
⎠
1
√
√ 0
2
2
' ( % X = LX !
⎧
1
1
⎪
x = − √ X + √ Y ,
⎪
⎪
⎨
2
2
y = Z
⎪
⎪
1
1
⎪
⎩ z = √ X + √ Y ,
2
2
&
4(X −
√
2)2 + 8Y
2
− 2(Z + 1)2 − 5 = 0.
⎧
√
⎨ X = X − 2,
Y = Y ,
⎩
Z = Z + 1,
4X 2 + 8Y 2 − 2Z 2 − 5 = 0 ⇔
2
2
2
Y
Z
X
+
−
= 1.
5/4 5/8 5/2
!
2x2 + 3y2 − 4x + 6y − 6z − 7 = 0
"#
$
%
2x2 − 4x + 3y 2 + 6y − 6z − 7 = 0 ⇔
⇔ 2(x2 − 2x + 1) + 3(y 2 + 2y + 1) − 6z − 12 = 0 ⇔
⇔z+2=
$
(x − 1)2 (y + 1)2
+
.
3
2
&
X = x − 1,
Y = y + 1,
' & &
2
Z=
(
2
2
Y
X
+
.
3
2
)
*
2
Z = z + 2.
2
2x + 5y + 2z − 2xy − 4zx + 2yz + 2x − 10y − 2z − 1 = 0,
⎛
⎞
2 −1 −2
1 ⎠
A = ⎝ −1 5
−2 1
2
2 − λ −1
−2
−1 5 − λ
1
−2
1
2−λ
=0
! "! # $ % !% &% ! '
!% # !# ! !'
((
−λ −1
−2
0 5−λ
1
−λ
1
2−λ
−2λ
0
λ
=0⇔ 0
5
−
λ
1
−λ
1
2−λ
⇔
−λ(λ2 − 9λ + 18) = 0 ⇔ λ(λ − 3)(λ − 6) = 0
λ1 = 6,
)
λ2 = 3,
λ3 = 0.
⎧
1
⎪
⎧
⎪
α11 = − √ ,
⎪
(2
−
3)α
−
1α
−
2α
=
0,
⎪
⎪
11
21
31
6
⎪
⎪
⎨
⎨
2
−1α11 + (5 − 3)α21 + 1α31 = 0,
α21 = √ ,
⇒
−2α11 + 1α21 + (2 − 3)α31 = 0,
⎪
⎪
6
⎪
⎪
⎪
⎩ α2 + α2 + α2 = 1
1
⎪
⎪
11
21
31
⎩ α31 = √ .
6
⎧
1
⎧
⎪
⎪
α12 = − √ ,
⎪
(2
−
6)α
−
1α
−
2α
=
0,
⎪
⎪
12
22
32
3
⎪
⎪
⎨
⎨
1
−1α12 + (5 − 6)α22 + 1α32 = 0,
α22 = − √ ,
⇒
⎪
⎪
12 + 1α22 + (2 − 6)α32 = 0,
3
⎪ −2α
⎪
⎩
⎪
2
2
2
1
⎪
+ α22
+ α32
=1
α12
⎪
⎩ α32 = √
3
⎧
⎧
1
⎪
2α13 − 1α23 − 2α33 = 0,
α =√ ,
⎪
⎪
⎪
⎪
⎨ 13
⎨
2
−1α13 + 5α23 + 1α33 = 0,
α23 = 0,
⇒
−2α13 + 1α23 + 2α33 = 0,
⎪
⎪
⎪
⎪
⎪ α = √1 .
⎩ α2 + α2 + α2 = 1
⎩
33
13
23
33
2
! " # $% L
&
⎛
⎞
1
1
−√ −√
⎜
6
3
⎜ 2
1
⎜ √
√
−
L=⎜
⎜
6
3
⎝ 1
1
√
√
6
3
1
√
2
0
⎟
⎟
⎟
⎟.
⎟
1 ⎠
√
2
' $% ! # X = LX
⎧
1
1
1
⎪
⎪
x = − √ X − √ Y + √ Z ,
⎪
⎪
6
3
2
⎪
⎨
2
1
y = √ X − √ Y ,
⎪
6
3
⎪
⎪
1
1
1
⎪
⎪
⎩ z = √ X + √ Y + √ Z ,
6
3
2
% ( ) #
! " &
! !
2X 2 + Y 2 − 2 = 0 ⇔ X 2 +
Y2
= 1.
2
x2 − 2y 2 + z 2 + 4xy − 8zx − 4yz − 14x − 4y + 14z + 16 = 0,
& *" +, ! # -
# &
⎛
⎞
1
2 −4
A = ⎝ 2 −2 −2 ⎠ .
−4 −2 1
! !
1−λ
2
−4
2
−2
−
λ
−2
−4
−2
1−λ
= 0,
λ1 = −3,
λ3 = −3.
λ2 = 6,
! "
# $ % & #
L
⎛
⎞
1
2
4
√
−√ −
⎜
3
3 5 ⎟
⎜ 25
1
2 ⎟
⎜ √
√ ⎟
−
L=⎜
⎟.
3 3√ 5 ⎟
⎜
5
⎝
5 ⎠
2
0
3
3
' # L
⎧
X
2
4
⎪
⎪
x = − √ − Y + √ Z ,
⎪
⎪
3
5
3 5
⎪
⎨
2
2
2
y = √ X + Y + √ Z ,
⎪
3
5
3
5
⎪
√
⎪
⎪
⎪
⎩ z = 2 Y + 5 Z ,
3
3
# $ X ! Y ! Z
# ! ("
# X Y ! X Z ! Y Z )
* #
& $ ! # $ $
$
2
Y
+ Z 2 = 0.
1/2
+ ! * det A = 54, det D = 0
X 2 − 2Y 2 + Z 2 = 0 ⇔ X 2 −
) # #
$ $
# &
, x2 + 2x + 2z2 − 4z + 4y2 + 2 = 0.
- x2 − 2x + y2 + 2y + z2 − 1 = 0.
. x2 + 4z2 − 2y2 − 4 = 0.
3y2 − 6y − x2 + 3z2 = 0.
3y2 − 6z2 − 2x2 − 6 = 0.
/ z2 + 2z − 3x2 + 6x − 3y2 − 5 = 0.
01 3x2 + 2z2 − 6y = 0.
x2 + 9y 2 + 18y + 9z + 9 = 0.
3x2 − 2z 2 − 6y = 0.
x2 +y 2 +5z 2 −6xy +2zx−2yz −4x+8y −12z +14 = 0
4x2 + 5y 2 + 6z 2 − 4xy + 4yz + 4x + 6y + 4z − 27 = 0.
x2 + 2y 2 + 3z 2 + 2x − 4y − 12z + 9 = 0.
y 2 + 2y + z 2 − 2z − 4x + 2 = 0.
! "! # $"
% $" & % $" $
' &
$$"
! ! ( $") *
$ + # &" δ $"$" " x0 , " $
" " x0 (x0 − δ; x0 + δ) - &"
&
. δ
P0 (x0 ; y0 )
δ(P0 ) = P (x; y)| (x − x0 )2 + (y − y0 )2 < δ
δ
/0# & " P 1"! δ $"$" "$& " " P0 $
$"& )' δ
P0 (x0 ; y0 )
!
ε
!
#
.
b
P (x; y)
P0 !
|f (x; y) − b| < ε
!
"
lim f (x; y) = b
x→x0
y→y0
z = f (x; y) P → P0
δ
lim f (P ) = b.
P →P0
Plim
→P
0
∃(ε > 0) ∀(δ(P0 ))∃(P ∈ δ(P0 ),
f (P ) = b
P = P0 ) ⇒ |f (P ) − b| < ε.
! "! #
! !
$ % & !' ! (
! % & !' !
!$( % %
) *+, *, !
lim f (x; y) = b ⇔ δ→0
lim f (x; y) = b δ = (x − x0 )2 + (y − y0 )2 %! P P0 ./ !
"! # !& & !
*,
b
x→x0
y→y0
z = f (x; y) P → P0
P0
P0 δ→0
lim f (x; y) = b
δ=
(x − x0 )2 + (y − y0 )2
+ ! lim
x→0
y→0
x2 + y 2
x2 + y 2 + 4 − 2
0
1 2 x0 = 0
⇒δ=
x2 + y 2
lim
x→0
y→0
= lim
δ→0
δ2
δ2
=
= lim √
x2 + y 2 + 4 − 2 δ→0 δ 2 + 4 − 2
x2 + y 2
√
√
δ2 + 4 + 2
2 + 4 + 2 = 4.
=
lim
δ
δ→0
δ2 + 4 − 4
x2 + y 2
x2 + y 2 + 4 − 2
P → P0
P0 (0; 0)
y0 = 0 P0 (0; 0) ⇒
x→x
lim f (x; y)
0
y→y0
lim
x→x0
lim f (x; y)
y→y0
lim
y→y0
lim f (x; y) ,
x→x0
!
" #
$% & x→x
lim f (x; y) ∀y ∈ δ
0
y0,
y→y0
y = y0 ∃ lim f (x; y)
∀x ∈ δ x0,
x→x
∃ lim f (x; y) ∃ lim lim f (x; y) ∃ lim lim f (x; y)
y→y
y→y x→x
x→x y→y
lim lim f (x; y) = lim lim f (x; y) = lim f (x; y)
y→y x→x
x→x y→y
x = x0
$% '
0
0
0
0
0
0
0
0
0
x→x0
y→y0
0
( " )
lim lim
x2 + y 2
= lim
=
x2 + y 2 + 4 − 2 y→0 y 2 + 4 − 2
y2
y2 + 4 + 2
2 + 4 + 2 = 4.
=
lim
y
= lim
y→0
y→0
y2 + 4 − 4
x2 + y 2
x→0
lim lim
= 4
2
y→0
x + y2 + 4 − 2
y→0 x→0
*
y2
+
" , - ./
x→x
lim lim
x→x0 +0
0 −0
! lim f (x) #
x→x0
0 2x lim
f (x; y) !
x→0
y = f (x)
y→0
P (x; y) P0 (x0; y0 )
2
2
y
$% $ ! lim xx2 −
+ y2
x→0
y→0
y = λx
x2 − y 2
x2 − λ2 x2
y = λx lim 2 2 = x→0
lim 2
=
2 2
x→0
y→0
2
=
x +y
x +λ x
1−λ
1 + λ2
λ
!" #
!" $ f (x; y)
P → P0
%&
' ( &
)
lim f (P ) = b ⇔ lim (f (P ) − b) = 0 ⇔ f (P ) − b
P →P0
P →P0
P → P0
* )
+, ! )
!" - n u = f (P )
P0
Plim
f (P ) = f (P0 )
→P
!" . P0 u = f (P )
0
/ ) &
& & ) +
&
0 +, +
' )
' +,
1
!" 2 ! n f1(P ) f2(P )
P0
f1 (P ) + f2 (P )
f1(P )/f2(P )
"
# f1(P ) − f2(P ) f1(P ) · f2(P )
f2(P0) = 0
3 4
')
, +,
'
P0
f (P )
z = x − 1y + 1
x − y + 1 = 0
! y = x + 1 ! " ! # $
% &' ( ! ! ) *
+ * #* y = x + 1
, - # ! %
! $
! ! * . " ! #$
%
) / / " !
* ! #
z = f (P )
! D
" ! f (P )
P0 # P ! P → P0
! D
$ z = f (P )
! D %
012 % ∃ N > 0 : |f (P )| N ∀P ∈ D&
032
! '( m '( M
% ∃ P1 ∈ D : f (P1) = m ∃ P2 ∈ D : f (P2) = M &
0 2 # ! m M !
)
% ∀ c ∈ [m; M] ∃ P0 : f (P0) = c
4 z = 1 − x2 − y2 2 2
! D = {(x; y)|x + y 1} *
! O(0; 0) +
+ " | 1 − x2 − y 2 | 1 x2 + y 2 1
5 ) * m = 0 " / ' x2 +
+ y 2 = 1 6 " # #) * M = 1 "$
6 ! #
fy (x0 ; y0 ) = lim
Δy→0
Δy z
.
Δy
P (x; y)
z = f (x; y) ! " P (x; y)
# x y
$
% % &
fx (x; y), fy (x; y)
zx , zy
∂z ∂z
,
.
∂x ∂y
$ n
n > 2 % % ! '
( u = f (x; y; z)
x P0 (x0 ; y0 ; z0 ) x Δx
" ! &
Δx u = f (x0 + Δx; y0 ; z0 ) − f (x0 ; y0 ; z0 ).
$ u = f (x; y; z) ! x
P0 (x0; y0; z0)
ux (x0 ; y0 ; z0 ) = lim
Δx→0
Δx u
.
Δx
"
) !
% "
* +" "
# !
" ! %
,-.
z =
x2 − y 2
P0 (5; 3).
∂z
2x
x
=
=
;
∂x
2 x2 − y 2
x2 + y 2
∂z
5
5
|P = √
= ;
∂x 0
4
52 − 32
−2y
y
∂z
=
= −
;
2
2
2
∂y
2 x −y
x − y2
∂z
5
|P =
∂x 0
4
∂z
3
|P = − .
∂y 0
4
∂z
3
|P = − .
∂y 0
4
∂z
∂x
z = f (x; y) ! "
z = f (x; y) #
P0 (x0 ; y0 ) Oxy $% $ M0 (x0 ; y0 ; z0 )
& '() $ $ AM0B "
#$ y = y0 *
$ + #
z = f (x; y0) y = y0 , "
df (x; y0 )
= tg α"
"
dx
α - #$ O1X " " +" #$ Ox #"
AM0 B M0 . "
∂z
df (x; y0 )
f (x0 + Δx; y0 ) − f (x0 ; y0 )
=
.
=
lim
dx x=x0 Δx→0
Δx
∂x P0
∂z
$ "
= tg α /"
∂x P0
∂z
P0 (x0 ; y0 )
∂x
Ox
M0 (x0 ; y0 ; z0 )
z = f (x; y) y = y0
0 $
∂z
1
∂x
∂z
∂y
2 # $
+ * " $ #"
# " $
z
Z
y
01
0
y0
M0
αA
B
P0
X
x
∂z
∂x
z = f (x; y)
! "
∂z
∂z
∂
∂ 2z
∂ 2z
∂x
∂x
=
=
= fxy
=
f
(x; y);
2 (x; y);
x
∂x
∂x2
∂y
∂x∂y
∂z
∂z
∂
∂
∂ 2z
∂ 2z
∂y
∂x
=
= fyx
= 2 = fy2 (x; y).
(x; y);
∂x
∂y∂x
∂y
∂y
# u = f (x; y; z) $ %
∂
"
∂
∂u
∂x
∂x
∂
=
∂u
∂x
∂y
∂ 2u
∂ 2u
=
= fxy
= fx2 (x; y; z);
(x; y; z);
2
∂x
∂x∂y
∂u
∂
∂ 2u
∂x
=
= fxz
(x; y; z)
∂z
∂x∂z
&
% % "
n %
(n − 1) '
3
∂ z
∂x∂y
2
z = f (x; y) y
∂ 2z
∂x∂y
∂ 3z
=
∂x∂y 2
∂
∂ 2z
∂x∂y
.
∂y
∂ 2z
∂ 2z
∂ 3z
,
,
∂x∂y ∂y∂x ∂x∂y 2
! "
" z = f (x; y)
#$ %
z = x2y3
& "
∂z
∂z
= 2xy 3 ,
= 3x2 y 2 .
∂x
∂y
' "
∂z
∂ 2z
∂x
=
= 2xy 3 y = 6xy 2 ,
∂x∂y
∂y
∂z
∂
∂ 2z
∂y
=
= 3x2 y 2 x = 6xy 2 .
∂y∂x
∂x
∂
(
∂ 2z
∂y∂x !) *
!
*
∂ 2z
∂x∂y
s
t
∂x
t
= 2cos s ln 2 − sin
∂t
s
t
∂x
= 2cos s
∂s
∂x
∂t
t
1
t 1
= −2cos s ln 2 sin · .
s
s s
∂x
t
∂s
t
t
t
t t
− 2 = 2cos s ln 2 sin · 2 .
ln 2 − sin
s
s
s s
·
!" "
#$%& z = 2x2 − 3y2 − 2xy + 3x − 5y + 1.
2y − 3x
.
#$%' z = 2x
− 5y
#$%$ z = √yx− x .
#$%( z = √x − y.
#$%)* z = y −3 x .
3
x
z = 2y .
#$%))
#$%)+ z = sin
x
√
2 y
√
y
√
.
3
3 x
#$%)# z = tg
#$%), z = arcsin(3y − 2x).
#$%)- z = arctg 3x − 2√y .
#$%)& u = (y)xz .
#$%)' u = xyz .
y = ln(cos u − sin v).
u = (yz)x .
!
z = x2 + y 2 .
z = log3 (x − y 2 ).
" # $! % " $$ % " &
' # ( $$ % ) $
$ % '
$! %
" '
# $! %
*
! ' $
+
, -
#
! $! %
. !
"! $! % !
z = f (x; y) " '
. ! x y ! ( # Δx
Δy - $! % z = f (x; y) ! # Δz
!(# $ ! /
Δz = f (x + Δx; y + Δy) − f (x; y).
+ ,
0 # $! %
Δz #
( $ $! % z = f (x; y)
P (x; y) ! P1 (x + Δx; y + Δy) + 1,
2
*
z = xy2
/ 3 ! $ ! ! + , !
Δz = (x + Δx)(y + Δy)2 − xy 2 = xy 2 + y 2 Δx + 2xyΔy+
+2yΔxΔy + x(Δy)2 + Δx(Δy)2 − xy 2 = (y 2 Δx + 2xyΔy)+
+(2yΔxΔy + x(Δy)2 + Δx(Δy)2).
z
M(x;y;z)
Δz
N
0
M0(x0;y0;z 0 )
Δx
x
P0 (x0;y0)
y
P(x;y)
Δx
Δy
Δz
y Δx+
Δx Δy
+ 2xyΔy
2yΔxΔy + x(Δy) + Δx(Δy)
Δx Δy ! " " #
Δx → 0 Δy → 0 ! "
#
2
2
2
$ % "
" dy
" "
& ' dx " " (
# dx = Δx dy "' '
dy = f (x)dx ) ' "
x z = f (x; y)
' y d z = f (x; y)dx
y * " x + d z = f (x; y)dy
,- .
y = f (x)
x
x
y
z = f (x; y)
y
x y
dz = fx (x; y)dx + fy (x; y)dy.
Δz = f (x+Δx; y +Δy)−f (x; y)!
" Δx = dx Δy = dy # " "!
Δz ""
dz
$ " ω(Δx; Δy) $ ! "
ρ = Δx2 + Δy 2 "% " P (x; y) P1 (x + Δx; y + Δy)&
Δz = dz + ω(Δx; Δy),
lim
ρ→0
ω(Δx; Δy)
= 0.
ρ
P (x; y)
' (
P (x; y)
(
)!
"! " *
fx (x; y)
fy (x; y)
z = f (x; y)
+% (! $ % ! $ !
! (
(
$
(
,
! %!
!
" .
"!
, (
-
)
!
" "! " "
"
"
∂z ∂z
∂x ∂y
z =
P (x; y)
P (x; y)
!"
= f (x; y)
/ (
"
, 2 !
0, "
,
"!
u = f (x; y; z)
0, $ 1
"
Δu
%
"
Δu =
∂u
∂u
∂u
Δx +
Δy +
Δz + ω(Δx; Δy; Δz)
∂x
∂y
∂z
3
ω
lim = 0 ρ = Δx2 + Δy 2 + Δz 2
ρ→0 ρ
du =
∂u
∂u
∂u
dx +
dy +
dz.
∂x
∂y
∂z
z
= xy 2
∂z
∂z
dx+ dy !
∂x
∂y
∂z ∂z
" # $ #
% #
∂x ∂y
∂z
∂z
= (xy 2 )x = y 2 ,
= (xy 2 )y = 2xy.
∂x
∂y
dz =
& " " $ ''('
)' ) Oxy *
* ) ( ! " dz = y 2 dx + 2xydy +
* , " -( ! ' ) ./
- )' z = f (x; y) ") P0 (x0 ; y0 )
dz = fx (x0; y0 )Δx + fy (x0 ; y0)Δy dz = fx (x0; y0 )(x − x0 ) +
+fy (x0 ; y0 )(y − y0 )
0 (! ) 1 ) $ " ) -
2
) M0 M1 KM2
z − z0 = fx (x0 ; y0 )(x − x0 ) + fy (x0 ; y0 )(y − y0 ),
3 z 4
) ") K ) - ) )-)
" *# ( 1 - # "
5 ) 1 $ ) # #
! (
) ) - ) KN
2 0 *
$ )(" ' 3 " )
6 "
!
) ) - ) KN - 3 ' " ! ' ) MN 2
z
M(x;y;z)
K
dz
N
M2
0
M0(x0;y0;z 0 )
M1
Δx
x
P0 (x0;y0)
P(x;y)
y
Δx
Δy
Δz
z = f (x; y)
Δz = fx (x; y)Δx + fy (x; y)Δy + ω(Δx; Δy).
! ω(Δx; Δy) " # ρ = (Δx)2 + (Δy)2
$ ρ# Δx Δy# % ω(Δx; Δy)
&
'() *+
Δz ≈ fx (x; y)Δx + fy (x; y)Δy,
, z = f (x; y)#
Δz = f (x + Δx; y + Δy) − f (x; y).
$ Δz '() *+#
f (x + Δx; y + Δy) − f (x; y) ≈ fx (x; y)Δx + fy (x; y)Δy,
f (x + Δx; y + Δy) ≈ f (x; y) + fx (x; y)Δx + fy (x; y)Δy.
! ! "# !!! ! P (x+Δx; y +Δy)$
! P (x; y)$ ! ! ! "# !
! P (x; y)
% & ! " ! "# n !!!
n > 2 '!$ n = 3
f (x + Δx; y + Δy; Δz + Δz) ≈
(
≈ f (x; y; z) + fx (x; y; z)Δx + fy (x; y; z)Δy + fz (x; y; z)Δz.
√
1
.
2.952 + 4.012
1
) ! * ! !+ ) "#, z =
-!
x2 + y 2
"
. "# +
1
1
≈
+ zx Δx + zy Δy.
2
2
2
(x + Δx) + (y + Δy)
x + y2
'! ! !+
x
y
; zy = − 2
.
zx = − 2
2
3/2
(x + y )
(x + y 2 )3/2
- !! x = 3$ Δx = −0,05$ y = 4$ Δy = 0,01 - +
1
4
3
1
≈ + √ 0, 05 − √ 0,01 ≈ 0,21.
2
2
5 5 5
5 5
2, 95 + 4, 01
/!$ ! . !! *! ,
1
0, $
≈ 0,201 1
2,952 + 4,012
! ,! !!!! ""!!# &
0! "#
2 z = f (x; y) 3 "# ! !!! x y $ !!
!! ! ! ! & $
""!!# & ""!!#$ ! ""!!
# & + d(dz) = d2 z
dx dy
x y
∂ ∂z
∂z
dx +
dy dx+
∂x ∂x
∂y
∂z
∂ ∂z
dx +
dy dy =
+
∂y ∂x
∂y
∂ 2z
∂ 2z 2
∂ 2z
∂ 2z 2
dxdy
+
dxdy
+
dx
+
dy .
=
∂x2
∂x∂y
∂y∂x
∂y 2
d2 z = d(dz) = d
∂z
∂z
dx +
dy
∂x
∂y
=
! " # #
∂ 2z 2
∂ 2z 2
∂ 2z
dxdy
+
d2 z =
dx
+
2
dy .
∂x2
∂x∂y
∂y 2
$ %# & ' % (
'
)
d3 z =
∂ 3z 3
∂ 3z
∂ 3z
∂ 3z 3
2
2
dx
dx
+
3
dy
+
3
dxdy
+
dy .
∂x3
∂x2∂y
∂x∂y 2
∂y 3
* % % # "+ ' n(%
'
&
" ,-
n
∂ nz
Cni i n−i dxi dy n−i .
dn z =
./
∂x ∂y
i=0
' z = f (x; y) #0 %(
) ' - '
(
t : x = x(t), y = y(t) 1% z & '
t # - )
∂z
∂z
& ' dz
#
(
dt
∂x
∂y
dx
dy
2 ) # " % #
dt
dt
' x = x(t) y = y(t) - # t '
z = f (x; y) -+ # (x; y) (
'
t # + Δt3 %
x y # - + Δx Δy
' z 4 + Δz 1 ' z & -
Δz
∂z
∂z
Δx +
Δy + ω(Δx; Δy),
∂x
∂y
ω
! lim = 0 " ρ = Δx2 + Δy 2 # $ !
ρ→0 ρ
Δt % & ' Δt → 0 !
Δz =
lim
Δt→0
Δz
Δx ∂z
Δy
ω
∂z
=
lim
+
lim
+ lim
.
Δt→0
Δt→0
Δt→0
Δt
∂x
Δt
∂y
Δt
Δt
(
) ' * $ &% * ! + "
&* * ! + "
dz
Δx
dx
Δy
dy
=
=
$ & , lim
lim
Δt→0 Δt
dt
dt Δt→0 Δt
dt
, *
ω
ω ρ
= lim
·
lim
Δt→0 Δt
Δt→0
ρ Δt
#
= lim
Δt→0
ω
ρ
· lim
.
ρ Δt→0 Δt
!
Δx2 + Δy 2
ρ
= lim
=
Δt→0 Δt
Δt→0
Δt
2
2
2
Δx
dx
Δy
= lim
+
=
+
Δt→0
Δt
Δt
dt
lim
-
dy
dt
2
.
dx
dt
Δt → 0
' ' ' $
dy
ω
. !
' $ ! lim = 0
!
Δt→0 ρ
dt
ω
ω
' ρ → 0 ,
" lim = lim = 0
ρ→0 ρ
Δt→0 ρ
2
2
ω
dx
dy
=0·
lim
+
= 0.
Δt→0 Δt
dt
dt
/! & +
(
$
dz
∂z dx ∂z dy
=
+
.
dt
∂x dt
∂y dt
3
y= t
dz
dt
z = y x x = cos t
dz
dy
dx
= (y x ) ·
+ (y x )y
= −y x ln y sin t + 3t2 xy x−1 =
dt
dt
dt
= −t3 cos t ln t3 sin t + 3t2 cos t · t3(cos t−1) = t3 cos t−1 (3 cos t − 3 ln t sin t) =
= t(sin t)t
2 −1
(t cos t + 2 sin t · ln sin t).
z = f (x; y) ! y = y(x)
"# z #$ $ x : z = f (x; y(x))
% $ !# #&#' ( $ t )*
x +
dz
∂z dx ∂z dy
=
+
.
dx
∂x dx ∂y dx
,
dx
= 1
dx
-
dz
∂z ∂z dy
=
+
.
dx
∂x ∂y dx
. !$
!$ / - $ & !#& z
∂z
x 0# /
1 !# #!/ *
∂x
&/ z = f (x; y) /# 2& y !
dz
x . ( !#
' ! !$ *
dx
& !# 3$ #$ $
z = f (x; y(x)) % !# & 2# &! $ *
!#$
+# 3 z = f (x; y) ( x = x(u; v)
y = y(u; v) 4)# z 3 #!/ !
&/ *
∂z
∂z
&/ u v , $# & !#&
- $ 3$
∂u
∂v
F (x; y)
x
M (x ∈ M)
y x
M
! y = ϕ(x)
y = ϕ(x)
! "! !#"!
F (x; ϕ(x)) = 0,
"
" x "!
M $!
% ! & !
y = f (x) '
( !" !) y !"
% " ""! * + #
( !) !" !) y ,
y = log3 (x3 + 1).
-.
/!
0!" $! ! # "
!'
+ 1 !#"!
! ! 2 + % " "!
"!'
( + "! y # -. &
3log3 (x
3 +1)
− x3 − 1 = x3 + 1 − x3 − 1 = 0.
% ! "&
# & 2 x ∈ M "! !"! !
") & y ! 23
" "! " x '
2 $! ! ")
x2 + y 2 − 1 = 0 '
!
! # "!)
(
x2 + y 2 − 1 = 0 !" !) y ,
y=
√
1 − x2 ,
√
y = − 1 − x2 .
4 "! !) &! " 2
"! !)
$!
2 2 #
4 '
3y − 3y + x3 − 1 = 0
! 2 2 y ! "3"! 2! & x
y ! 23 2 x = 0 y = 05
x=1 y=1
F (x; y) = 0
x2 + y 2 + 1 = 0
x y
!
y
x
"#
! #
" "
$
y%
F (x; y) = 0
&
F (x; y)
!
!
'
#
#
'
"
()* F (x; y)
Fx (x; y) Fy (x; y)
P0 (x0; y0) F (x0; y0) = 0 Fy (x0; y0) = 0
F (x; y) P0 (x0; y0)
y = y(x)
x0
y(x0 ) = y0
+
, " !
,
!
/
-().)
" #
!
y = y(x) " P0 (x0 ; y0 ) #
F (x; y(x)) ≡ 0 x
/ !
! !
dF
=0
dx
"
-().0
dF
∂F
∂F dy
=
+
dx
∂x
∂y dx
!
∂F
∂F dy
+
=0
∂x
∂y dx
∂F
dy
= − ∂x .
∂F
dx
∂y
,
-()*.
" 1 ! " "
! "
y x3 − 3x + y2 − xy − 1 = 0
P (1; 1)
F (x; y) = x2 −2x+3y 2 −xy−1 = 0
∂F
∂F
= 3x2 −3−y
= 2y−x !"
∂x
∂y
∂F
dy
3x2 − 3 − y
3x2 − 3 − y
= − ∂x = −
=
.
∂F
dx
2y − x
x − 2y
∂y
# $ % & ' x
y =
(
(6x − y )(x − 2y) − (1 − 2y )(3x2 − 3 − y)
.
(x − 2y)2
# (
) & ' * y
y =
(++ ,
(3x2 − 12xy + 3 + y)(x − 2y) − (x − 6x2 + 6)(3x2 − 3 − y)
.
(x − 2y)3
((
+ P (1; 1)
dy
= −6.
x=1 = 1, y | x=1
y=1
dx y=1
- & '* y *
x & (
)
+ +(* + + ( (. Ox 45o tg ϕ = 1"
/+ ( * $ +$ ) ) y =
= f (x) ( ( & 0 & ,
' * $ dy = f (x)dx ((* & (
* *(* x ( ) ) +$ ) +,
) ) x = ϕ(t)
1* +$
(++ 2 &2 u = f (x; y; z; . . . ; t) ( ,
' &)
$ +$
n &2 u = f (x; y; z; . . . ; t) (2 * ( .
∂u
∂u
∂u
∂u
dx +
dy +
dz + · · · +
dt
du =
∂x
∂y
∂z
∂t
( * *.(* x y z . . . t ( & ,
& +$ * 2 &2
z = f (x; y) x y
! "
dz =
∂z
∂z
dx +
dy.
∂x
∂y
dz =
∂z
∂z
du +
dv.
∂u
∂v
# x y
" x = x(u; v) y = y(u; v) $%
z u v &
' ()* +,- ()* +.∂z
∂z ∂x ∂z ∂y
=
+
,
∂u
∂x ∂u ∂y ∂u
/
∂z
∂z ∂x ∂z ∂y
=
+
.
∂v
∂x ∂v ∂y ∂v
∂z
∂z ∂x ∂z ∂y
∂z ∂x ∂z ∂y
+
du +
=
+
dv =
dz =
∂x ∂u ∂y ∂u
∂v
∂x ∂v ∂y ∂v
∂z ∂x
∂z ∂y
∂z ∂y
∂z ∂x
du +
dv +
du +
dv =
=
∂x ∂u
∂x ∂v
∂y ∂u
∂y ∂v
∂z ∂x
∂x
∂z ∂y
∂y
∂z
∂z
=
du +
dv +
du +
dv =
dx +
dy,
∂x ∂u
∂v
∂y ∂u
∂v
∂x
∂y
∂y
∂y
du +
dv = dy.
0 ∂u
∂v
/ dz %
∂x
∂x
du +
dv = dx,
∂u
∂v
dz =
∂z
∂z
dx +
dy
∂x
∂y
x y
1 2 %
3
z = xy 2
dz M(2,00; 1,00) Δx = 0,20
Δz
Δy = 0,10
Δz = f (x + Δx; y + Δy) − f (x; y) = (x +
+Δx) · (y + Δy)2 − xy 2 x = 2 y = 1
M Δz = 2,20 · 1,102 − 2,00 · 1,002 ≈ 0,66
∂z
∂z
dz = dx + dy = y2Δx + 2xyΔy.
∂x
∂y
dx = Δx dy = Δy x y
M
dz = 1,002 · 0,20 + 2 · 2,00 · 1,00 · 0,10 = 0,60.
! Δz dz 0,06
ρ = Δx2 + Δy2 ≈ 0,22.
" Δz ≈ 0,66; dz = 0,60
#$ % dz z = x3y2
∂z
∂z
dx +
dy = 3x2 y 2 dx + 2x3 ydy.
∂x
∂y
dz = 3x2 y 2 dx + 2x3 ydy.
dz =
"
#$ #
1,012 · 0,983 .
&' ( )#$ *+
,, dz , z = x2y3 x = 1
y = 1 Δx = 1,01 − 1 = 0,01 Δy = 0,98 − 1 = −0,02 - .
1,012 · 0,983 f (x + Δx; y + Δy)
dz =
∂z
∂z
dx +
dy = 2xy 3 Δx + 3x2 y 2 Δy =
∂x
∂y
= 2 · 1 · 13 · 0,01 + 3 · 12 · 12 (−0,02) = −0,04.
- (!/ Δz ≈ dz
f (x + Δx; y + Δy) − f (x; y) ≈ dz ⇒ f (x + Δx; y + Δy) ≈ f (x; y) + dz
0 f (x; y) = 12 · 13 = 1
1,012 · 0,983 ≈ 1 − 0,04 = 0,96.
∂z dz
∂x dx
x
z = 2y
y = sin x
∂z
∂x
dz
! "
dx
x
∂z
1
= 2 y ln 2 ·
∂x
y
x
x
dz
1
x
= 2 y ln 2 + 2 y ln 2 − 2 cos x =
dx
y
y
x
x
ln 2
x
sin
2 x
1
x
= 2 sin x ln 2
cos x =
+ 2 sin x ln 2 − 2
(1 − x ctg x) .
sin x
sin x
sin x
dy
dx
#
y = xy
$ y = xy F (x, y) =
dy
yxy−1
= xy − y = 0 % & '()
=− y
dx
x ln x − 1
* " + ! , y
- " x ! y
.
/
" dz
Δx Δy
.
"
z = x2 y 2 ;
/ z = x2 y;
(01 (
(0 z =
Δz
" M(x; y)
M(1; 2);
M(1; 1);
Δx = 0,05;
Δx = 0,01;
Δy = 0,10.
Δy = 0,02.
"
x2 + y 2
.
x2 − y 2
dz
x
z = ln tg .
y
u = xyz.
z = xy x .
z = xey + yex .
z = ey sin x.
z = exy .
! " # $ %& ' "
$ ' #(' ( ) #
# " # $ ( ' " * + #
' (
sin 31◦ cos 61◦ .
! 0,993 · 1,022 .
√
,
3,98 + 2,95.
# -*
.
z = cos t
√
dz
dx
z = s − t t = tg x s = √x
3
√
du
u = xyz x = t + 2 y = et
dt
∂z
dz
∂x
dx
z = yx y = arctg x
v = tg x
* " ' (
dz
z = xy y = arcsin t x = ln t
dt
dz
dx
z = sin(u · v)
dy
dx
y = x + log3 y.
u =
√
x
u P (x; y; z)
l
u
∂u
< 0 u
∂l
∂u
!
∂l
u
"
#
$ % & Δx Δy Δz
P # P1 P = Δl &
& ' ( )*+,
Δx = Δl cos α; Δy = Δl cos β; Δz = Δl cos γ.
(-. )+
/ u
0* (0* -+ ! & Δu P (x; y; z)
Δu = Fx (x; y; z)Δx + Fy (x; y; z)Δy + Fz (x; y; z)Δz + ω,
(-. 0+
! ω ρ =
ω
lim = 0
ρ→0
Δx2 + Δy 2 + Δz 2
ρ
z
γ
p
α
Δy
β
p1
Δz
Δx
l
0
y
x
Δl
Δx Δy Δz
1 &
Δu = Δl u ρ = Δl Δx Δy Δz
Δl u =
Fx (x; y; z)Δl cos α
+ Fy (x; y; z)Δl cos β + Fz (x; y; z)Δl cos γ + ω.
! Δl "# $
Δl → 0%
∂u
Δl u
= lim
= lim (Fx (x; y; z) cos α+
Δl→0 Δl
Δl→0
∂l
ω
.
+Fy (x; y; z) cos β + Fz (x; y; z) cos γ + lim
Δl→0 Δl
& Fx (x; y; z)% Fy (x; y; z)% Fz (x; y; z) # $'
ω
ω
= lim = 0%
# Δl% $ $$ Δl→0
lim
Δl ρ→0 ρ
∂u
= Fx (x; y; z) cos α + Fy (x; y; z) cos β + Fz (x; y; z) cos γ.
∂l
( '
% $
% % # u l
! $) $% %
=
% cos α = 1% cos β = 0% cos γ = 0 % *% ∂u
∂l
= Fx (x; y; z)
+
z = f (x; y)
!"
z = f (x; y) l
∂z
= fx (x; y) cos α + fy (x; y) cos β.
∂l
P0 (5; 3)
+ # z =
l = 3i + 4j
-
3
3
3
=√
= ;
|l|
5
32 + 42
' $ P0 '
cos α =
cos β =
,
x2 − y 2
4
4
= ,
|l|
5
'
' . / 0
,
∂z
3 3
3
53 34
−
= − = .
=
∂l P0 4 5 4 5
4 5
20
P0
|P > 0
z = x2 − y2 ∂z
∂l
0
|P
∂z
∂l
0
=
3
.
20
!"# P (x; y; z)
u = F (x; y; z)
Fx (x; y; z)i + Fy (x; y; z)j + Fz (x; y; z)k.
$ u = F (x; y; z)
grad F (x; y; z) grad F (P ) grad u %
&
grad F (x; y; z) = Fx (x; y; z) + Fy (x; y; z) + Fz (x; y; z),
'!"()
∂u
∂u
∂u
grad u =
+ + .
'!"*)
∂x
∂y
∂z
+ , P (x; y; z)
u = F (x; y; z)
- grad F (P )
!". u = x2 + y2 − z2
P0 (1; 1; 1)
/ 0 1 2 2 P0
∂u
= 2x;
∂x
∂u
= 2;
∂x P0
∂u
= −2z;
∂z
∂u
∂u
= 2y;
= 2;
∂y
∂y P0
∂u
= −2.
∂z P0
3 '!"*)
grad u = 2i + 2j − 2k.
grad u = 2i + 2j − 2k.
l = i cos α + j cos β + k cos γ
l
grad u
u
.
l grad u = ∂u
∂l
u = F (x; y; z)
! " #
$
# % ! $ & ' (
)
grad u = Fx (x; y; z)i + Fy (x; y; z)j + Fz (x; y; z)k.
'
%
!l grad u = grad u · l =
= Fx (x; y; z) cos α + Fy (x; y; z) cos β + Fz (x; y; z) cos γ =
$
∂u
,
∂l
∂u
+
∂l
% # # ! # u = F (x; y; z) ' % !
l %
% +
! " # grad u
# ! # u = F (x; y; z) !
l
, % ϕ %+$ $ %
% l grad u
- $ !l grad u = | grad u| cos ϕ ' % . %
* #
! $ # ! ! &
∂u
= | grad u| cos ϕ.
/
∂l
0 !
l grad u ! $ & (ϕ = 0)
!
∂u
% $ 1
$ # ! ! &
∂l
| grad u|
- % % % ! ( $ % $&2% $ grad u
, &$ $
grad u . " # ! # u = F (x; y; z)
!$# # % % ! %
%
$
% # . " # ! #
3 # % % ! + grad u = grad F (x; y; z) $
P (x0 ; y0 ; z0 ) ! ( # ! ( $#2 '
' ! ( % $
F (x; y; z) = C0
F (x; y; z) − C0 = 0.
z
grad F(P 0 )
r (t 0 )
90
p0
o
L
F(x;y;z)-C 0=0
0
y
x
L
P0 ! "
⎧
⎨ x = x(t),
y = y(t),
⎩ z = z(t),
# x(t)
y(t) z(t) $ %%& ' % & t ( x0 = x(t0 )
y0 = y(t0 ) z0 = z(t0 ) ) * + r = (x; y; z)
, + , %- r = r(t) . +
', ' (x(t0); y(t0); z (t0)) = r (t0)
+, , L r(t0)
/, + ,r̄ (t0 ) = lim
Δt→0
r̄(t0 + Δt) − r̄(t0 )
.
Δt
0 r̄(t0 + Δt) − r̄(t0) , , L
, P0 (r̄(t0)) # ", , r̄(t0 + Δt)1
r̄(t + Δt) − r̄(t0 )
0
2# Δt → 0
Δt+
2 , L ' x(t) y(t) z(t)
' ' +
+
L + ", 3
F (x(t); y(t); z(t)) − C0 = 0.
t
(C0)t = 0
∂F
∂F
∂F
x (t) +
y (t) +
z (t) = 0.
∂x
∂y
∂z
t = t0
Fx (x0 ; y0 ; z0 )x (t0 ) + Fy (x0 ; y0 ; z0 )y (t0 ) + Fz (x0 ; y0 ; z0 )z (t0 ) = 0.
grad u(P0 ) = Fx (x0 ; y0 ; z0 ) + Fy (x0 ; y0 ; z0 ) + Fz (x0 ; y0 ; z0 )
r (t0 ) = x (t0 )i + y (t0 )j + z (t0 )k,
L! "
#$%!&&'
grad u(P0) = 0! " #$%!&&' (
grad u(P0) r(t0 )
L P0!
" )
*+ !
grad u(P0 ) · r (t0 ) = 0.
u = F (x; y; z)
P0
grad F (P0 )
P0
) ) z = f (x; y) (
grad f (x; y) = fx (x; y) + fy (x; y).
#$%!&,'
∂z
- * ∂l
= | grad z| cos ϕ,
l grad z = ∂z
∂z
∂l
∂l
ϕ . l grad z!
/ (
z = f (x; y) grad f (x0; y0) (
P0 (x0; y0)!
F (x; y; z) = 0,
F (x; y; z) ! P0 (x0; y0; z0)
! " # $ $%
$ " & $ ! P0 " #' (
' ! ! P0 " # $ ) "
) grad F (P0 )
* $ ) F (x; y; z) = 0
! P0
grad F(P 0)
z
P0 (x0;y0 ;z0 )
y
x
+ #
" ' ) !
,)
- ! P0 .
A(x − x0 ) + B(y − y0 ) + C(z − z0 ) = 0.
/ ! N (A; B; C) & (
" $) ) 0 (
) .
Fx (x0 ; y0 ; z0 )(x − x0 ) + Fy (x0 ; y0 ; z0 )(y − y0 ) + Fz (x0 ; y0 ; z0 )(z − z0 ) = 0.
P0
P0 !"#$
x − x0
y − y0
z − z0
=
=
.
m
n
p
% & s(m; n; p) & '
( $
y − y0
z − z0
x − x0
=
=
.
")#
Fx (x0 ; y0 ; z0 )
Fy (x0 ; y0 ; z0 )
Fz (x0 ; y0 ; z0 )
!
z = x2 + 2y2 P0 (1; 1; 3)
* + $ ,+
+ 2y 2 − z = 0 , F (x; y; z) = x2 + 2y 2 − z
- grad F (P0 )$
Fx (x; y; z) = 2x;
Fy (x; y; z) = 4y;
"!#$ x2 +
Fz (x; y; z) = −1.
Fx (P0 ) = 2; Fy (P0 ) = 4; Fz (P0 ) = −1.
%
$
" #
2(x − 1) + 4(y − 1) − 1(z − 3) = 0,
$ 2x + 4y − z − 3 = 0.
% ")# $
x−1
y−1
z−1
=
=
.
2
4
−1
y−1
z−1
x−1
=
=
.
. $ 2x + 4y − z − 3 = 0
2
4
−1
* '
& z = f (x; y) !#
/ ( 0 (
,+
−f (x; y) + z = 0.
, F (x; y; z) = −f (x; y) + z
"1#
gradF (P0)
Fx (x0 ; y0 ; z0 ) = −fx (x0 ; y0 );
Fy (x0 ; y0 ; z0 ) = −fy (x0 ; y0 );
Fz (x0 ; y0 ; z0 ) = 1.
−fx (x0 ; y0 )(x − x0 ) − fy (x0 ; y0 )(y − y0 ) + (z − z0 ) = 0,
z − z0 = fx (x0 ; y0 )(x − x0 ) + fy (x0 ; y0 )(y − y0 ).
x − x0
y − y0
z − z0
=
=
.
−fx (x0 ; y0 )
−fy (x0 ; y0 )
1
!
" P0 # $!
% & ' ( ! # )
% #
F (x; y; z) = 0 "
Fy (P0 )
Fz (P0 )
Fx (P0 )
; cos β =
; cos γ =
,
| grad F (P0 )|
| grad F (P0 )|
| grad F (P0 )|
*
2
2
2
| grad F (P0 )| =
(Fx (P0 )) + (Fy (P0 )) + (Fz (P0 )) , #)
% z = f (x; y)- . !- " $ )
cos α =
+
−fy (x0 ; y0 )
1
−fx (x0 ; y0 )
; cos β =
; cos γ =
,
| grad F (P0 )|
| grad F (P0 )|
| grad F (P0 )|
/
| grad F (P0 )| = (fx (x0 ; y0 ))2 + (fy (x0 ; y0 ))2 + 1
cos α =
+
2
z = x2 +2y
o
M(1; 1)
Ox 60
0 1 ! cos α = cos 60o = 21 cos β = cos(90o − 60o ) = cos 30o =
√
3
2
cos β
cos2 α + cos2 β = 1
M
∂z
|M = 2x|x=1 = 2;
∂x
∂z
|M = 4y|y=1 = 4.
∂y
√
√
∂z
3
∂z
∂z
1
|M =
|M cos α +
|M cos β = 2 + 4
= 1 + 2 3 ≈ 4,46.
∂l
∂x
∂y
2
2
XOY M(1; 1)
!
" z = x2 + 2y2 # $ %
XOY
77o
&'( z = 2x2 +y2
M(1; 1)
O(0; 0)
) * +
MO +
, - .
" MO = (0 − 1; 0 − 1) = (−1; −1)
+
MO
.
|MO|
√
|MO| = (−1)2 + (−1)2 = 2
1
1
1
l MN = − √ ; − √
⇒ cos α = − √ ;
2
2
2
l MN =
1
cos β = − √ .
2
/ 0 +
∂z
∂z
|M = 4x|x=1 = 4;
|M = 2y|y=1 = 2 ⇒
∂x
∂y
1
1
6
∂z
√
√
|M = 4 −
⇒
+2 −
= − √ ≈ −4,3.
∂l
2
2
2
M
! " ! XOY
#$%
z = x2 + 2y2
M(1; 1)
& ' ( ) ! #$*
+ + ! grad z (
grad z|M =
∂z
∂z
|M ; |M
∂x
∂y
= (2; 4).
, - #$
! ' "
" .
∂z
∂x
2
+
2
∂z
/
∂y
M "
√ √
22 + 42 = 20 ≈ 4,47 " '
#$*
#$#
x2 y 2 z 2
+
−
=0
16
9
8
P0 (4; 3; 4)
& ' (
+
x2 y 2 z 2
+
−
F (x; y; z) = 0 "( F (x; y; z) =
16
Fx (x; y; z) =
x
;
8
1
Fx (P0 ) = ;
2
9
8
2y
z
; Fz (x; y; z) = − .
9
4
2
Fy (P0 ) = ; Fz (P0 ) = −1.
3
Fy (x; y; z) =
, %0*1 !
(
1
2
(x − 4) + (y − 3) − (z − 4) = 0
2
3
3x + 4y − 6z = 0.
x−4
y−3
z−4
=
=
.
1/2
2/3
−1
y−3
z−4
x−4
=
=
3x + 4y − 6z = 0
1/2
2/3
−1
! "#$ %
&# ' (!) *! Ox + α
, z = 2x2 + xy + 3y2 M(2; −2) α = −45o
- z = x2 − y2 M(−1; 1) α = 120o
z = xy
z =
M(2; 1) cos α =
1
0 < β < π/2
3
x2 − y 2 M(5; 3) cos α = cos β < 0
! "#$
&# M ' ) . &# # &# N
z = xy1 M(1; 1) N (−1; 1)
z = √x − y M(1; 0) N (0; 1)
u = xy + yz + zx M(2; 1; 3) N (5; 5; 15).
/ , + "#$ %
&# ! . "#$ &#
+ 0*1! #* ( "#$
&#
/ z = 2x3 + 3y3 − 2xy M(2; 1).
z = 2 y2 − x2 M(3; 5).
u = 2xyz M(3; 2; 1).
, u = 4x2 + y2 + z2 M(1; 1; 1).
-
z=
x2 y 2
−
2 3
P0 (2; 3; −1)
!
" # $ % & '
( ' )
* # + ( ! $
, -./0
f (x) = f (x0) +
+
(
f (x0 )
f (x0 )
(x − x0 ) +
(x − x0 )2 + · · · +
1!
2!
,1--/
f n−1 (x0 )
(x − x0 )n−1 + O((x − x0 )n ),
(n − 1)!
! # $ $ n ρ → 02 #
3 (# 2 4
(# 4 5 ( 2 ' " $
, ! / ( * ($$ 5 # 0
O(ρn )
ρ = x − x0
1
fx (x0 ; y0 )Δx + fy (x0 ; y0 )Δy +
1!
1
fxx (x0 ; y0 )Δx2 + 2fxy
+
(x0 ; y0 )ΔxΔy + fyy
(x0 ; y0 )Δy 2 + O(ρ3 ),
2!
ρ = Δx2 + Δy 2 Δx = x − x0 Δy = y − y0
,1-/
f (x; y) = f (x0 ; y0 ) +
(
2
2
6 " 2 # ! # $
$ 2 ' " # 0
f (x; y) = f (x0 ; y0 ) +
+
df (x0 ; y0 ) d2 f (x0 ; y0 )
+
+ ···+
1!
2!
dn−1 f (x0 ; y0 )
+ O(ρn ).
(n − 1)!
,1-7/
8 2 # ,7.7/ n = 2
O(ρn )
m
f (x1 ; x2 ; . . . ; xm ) = f (x01 ; x02 ; . . . ; x0m ) + fx 1 (x01 ; x02 ; . . . ; x0m)Δx1 + !"#"$
+fx 2 (x01 ; x02 ; . . . ; x0m )Δx2 + · · · + fx m (x01 ; x02 ; . . . ; x0m )Δxm + O(ρ2 ),
m
Δxi = xi − x0i i = 1, 2, . . . , m ρ =
Δx2i
i=1
% m m
fi (x1; . . . ; xm ) &
⎧
f (x ; x ; . . . ; xm ) = 0,
⎪
⎪
⎨ 1 1 2
f2 (x1 ; x2 ; . . . ; xm ) = 0,
.........
⎪
⎪
⎩ f (x ; x ; . . . ; x ) = 0.
m 1
2
m
"##
!"#'$
m P0 (a1 ; a2 ; . . . ; am )
!
(
X ! m
$ F
⎛
x1
⎛
⎞
X = ⎝ ⎠ ,
xm
F (X) = ⎝
f1 (x1 ; . . . ; xm )
⎞
⎛
⎠,
0 = ⎝ ⎠
0
fm (x1 ; . . . ; xm)
!"#'$
*
0
⎞
!"#)$
!"#+$
*, - !"#'$ . /
-
% *, - Xn !"#+$
ε -
F (X) = 0.
⎛
xn1
⎞
Xn = ⎝ ⎠ ,
xnm
⎛
ε=⎝
Δx1
Δxm
⎞
⎠,
!"#0$
Δxi = xi − xni xi = xni + Δxi
ε = X − Xn X = Xn + ε.
!
"#
⎧
⎨ f1 (xn1 + Δx1 , . . . , xnm + Δxm ) = 0,
...
⎩
fm (xn1 + Δx1 , . . . , xnm + Δxm ) = 0.
$ % & ' %()
*
' +)
, ) # %
' -
)
fi (xn1 + Δx1 , . . . , xnm + Δxm ) = fi (xn1 , . . . , xnm )+
+
m
fi,x
(xn1 , . . . , xnm )Δxj + α2 ρ2 ,
j
j=1
fi,x
(xn1 , . . . , xnm ) =
j
∂fi (xn1 , . . . , xnm )
∂xj
!
# . )
) / %0 Δxnj ,
(j = 1, 2, . . . , m) % /n & 0 xn1 , . . . , xnm # %
, Δxj
⎧
(xn1 , ..., xnm)Δxn1 + ... + f1,x
(xn1 , ..., xnm)Δxnm = 0,
⎨ f1 (xn1 , ..., xnm) + f1,x
m
1
...
⎩
(xn1 , ..., xnm)Δxn1 + ... + fm,x
(xn1 , ..., xnm)Δxnm = 0.
fm (xn1 , ..., xnm) + fm,x
m
1
.
1 * )
% Δxnj (j = 1, . . . , m)#
/(n + 1) & 0 ' 2
xn+1
= xn1 + Δxn1 , . . . , xn+1
= xnm = +Δxnm .
1
m
xnj
2
3 # )
(j = 1, . . . , m)
"# & # %
% Δxnj
'
+)
) xn+1
%
j
.#
) # (n + 2) # & ! (
& # % % % 4 %
) ,4
)
δ
|Δxnj | < δ j = 1, . . . , m
! "
# $%&'( )
* * * +,
, $%&-(
$%&'( !
F (Xn ) + W (Xn ) · εn = 0,
$%&&&(
" W (Xn ) . ) /
0 ) fi (x1 , . . . , xm )
(xn1 , . . . , xnm)
⎛
⎞
∂f1
∂f1
...
⎜ ∂x1
∂xm ⎟
⎜
⎟
W (X) = ⎜ . . . . . . . . . ⎟ .
$%&&1(
⎝ ∂fm
∂fm ⎠
...
∂x1
∂xm
# $%&&&( ) )
εn = −W −1 (Xn )F (Xn ),
$%&&2(
−1
" W (Xn ) . ) ) /
Xn
3 (n + 1) 0
Xn+1 = Xn + εn .
$%&&%(
%&&
x1, x2, x3 x, y, z
%&& !"
x2 + 4y 2 − 1 = 0
" δ = 0,01
y − x3 = 0
#
,
x1 = x x2 = y f1 (x1 ; x2 ) = x2 + 4y 2 − 1 f2 (x1 ; x2 ) =
= y − x3
∂f1
∂f1
∂f1
∂f1
= 2x,
= 8y,
=
=
x2 + 4y 2 − 1
∂x
∂x
∂x
∂y
1
2
;
F (X) =
∂f2
∂f2
∂f2
∂f2
y − x3
= −3x2 ,
= 1.
=
=
∂x1
∂x
∂x2
∂y
W (X) =
2x 8y
−3x2 1
.
!" #
F (X) W (X)
$ x0 y0 δ%
& #$ $
'(%(% )
2
y2
x2 + 4y2 − 1 = 0 * x1 + (0,5)
= 1
2
a = 1 b = 0,5% ) y − x3 = 0
y = x3% # #
$ + ,% -./
y
0,5
1
-1
x
-0.5
0 #$ 1 $
$ x0 = 0,502
y0 = 0,25%
& $
1 '(%(% ! #
1 %% - %
$
( 3 ,4 .5/6
⎞
1
4y
−
2
2
⎜ 2x + 24x y
x + 12x y ⎟
W −1 (X) = ⎝
⎠.
3x2
x
2x + 24x2 y x + 12x2 y
⎛
⎧
ε = −W −1 (X)F (X)
x2 + 4y 2 − 1 4y(y − x3 )
−
,
2x + 24x2 y
x + 12x2 y
2 2
2
3x (x + 4y − 1)
x(y − x3 )
⎪
⎪
+
.
⎩ Δy = −
2x + 24x2 y
x + 12x2 y
x0 = 0,50 y0 = 0,25
! " Δx0 = 0,30# Δy0 = 0,10
⎪
⎪
⎨ Δx = −
$% !
x1 = x0 + Δx0 = 0,8 y1 = y0 + Δy0 = 0,35
! ! &
% %
! " x1 y1 ! Δx1 =
= −0,084# Δy1 = 0,0014# x2 = x1 + Δx1 = 0,716# y2 = y1 + Δy1 = 0,3514
! ! &
|Δx1 | < 0,01
% %
|Δy1 | < 0,01
'"! % % ( % &!
|Δx0 | < 0,01
|Δy0 | < 0,01
i
xi
yi
Δxi
Δyi
) ) ) ) * ) ) ) )
) +) ) ) )+ ) ))
* ) , - ) ) ))++ ) )).
) ,), ) )
/ & % i = 2 |Δx2| < 0,01#
|Δy2 | < 0,01
0 x3 = 0,71# y3 = 0,35
0,01
* z = f (x; y)
M (x ; y ) ∈ D(f ) x = x
y = y
f (x ; y ) > f (x; y)
(x; y) (x ; y ) δ
! "
0
0
0
0
0
0
0
0
0
z = f (x; y)
M0 (x0 ; y0 ) ∈ D(f )
f (x0 ; y0 ) < f (x; y)
(x; y) (x0; y0)
! "
# "
$ % " &
" ' #
( & x = x0 + Δx y = y0 + Δy )
f (x; y) − f (x0 ; y0 ) = f (x0 + Δx; y0 + Δy) − f (x0 ; y0 ) = Δf (x0; y0 ).
* + Δf (x0 ; y0 ) < 0 " , , "' ,
, "
, f (x; y)
M(x0 ; y0 )
-* + Δf (x0 ; y0 ) > 0 " , , "' ,
, "
, f (x; y)
M(x0 ; y0 )
. " # #
"
,
!" #
z = f (x; y) ! $ x = x0 y = y0 %
$ $ $ z
&
$ !
&
$
" y " /
y = y0 0 f (x; y0 ) # "
x 0 " x = x0 1 *
∂z
∂x
x=x0
y=y0
∂z
∂y
x=x0
y=y0
z
M0
y
P0
x
M0
z
y
P0
x
!
" #
$ %&
& ' ( "
&
$
& & %& %
%
) " # "
∂z
= +2x
* " ' ( z = x2 − y 2 " %
∂x
∂z
= −2y % # "
x = 0 y = 0 + $
∂y
' ( "
%&
&
,! $ ' (
"
# &
&
" %
( %
-
f (x; y)
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
·
−
∂x2
∂y 2
∂ 2 f (x0 ; y0 )
∂x∂y
fyy
(x0 ; y0 ) < 0
f (x; y)
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
·
−
∂x2
∂y 2
∂ 2 f (x0 ; y0 )
∂x∂y
2
>0
∂ 2 f (x0 ; y0 )
< 0;
∂x2
>0
∂ 2 f (x0 ; y0 )
> 0;
∂x2
2
fyy
(x0 ; y0 ) > 0
f (x; y)
2
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
∂ 2 f (x0 ; y0 )
·
−
< 0;
∂x2
∂y 2
∂x∂y
2
∂ 2 f (x0 ; y0 ) ∂ 2 f (x0 ; y0 )
∂ 2 f (x0 ; y0 )
·
−
= 0
∂x2
∂y 2
∂x∂y
!
" # $ $ !
% & ' (
(x0 ; y0 ) $ fxx
(x0 ; y0 ) fyy
(x0 ; y0 ) # )!
2
* fxx (x0 ; y0 ) · fyy (x0 ; y0 ) − fxy
(x0 ; y0 ) > 0
2
fxy (x0 ; y0 )
< 0.
# fxx
(x0 ; y0 ) < 0 fyy
(x0 ; y0 ) <
(x ; y )
fxx
00
+
z = x3 + y 3 − 3xy !
' ()
! "
#
⎫
∂z
⎬
= 3x2 − 3y = 0, ⎪
∂x
∂z
⎭
= 3y 2 − 3x = 0. ⎪
∂y
$% !
x1 = 1,
y1 = 1
x2 = 0,
y2 = 0.
2
2
∂ z
∂ 2z
∂ z
=
−3,
=
6x,
= 6y.
∂x2
∂x∂y
∂y 2
M(1; 1)
∂ 2z
∂ 2z
∂ 2z
A=
=
6,
B
=
=
−3,
C
=
= 6,
∂x2 x=1
∂x∂y x=1
∂y 2 x=1
y=1
y=1
y=1
AC − B 2 = 36 − 9 = 27 > 0; A > 0( C > 0).
(1; 1) ! "
zmin = −1.
#
M2 (0; 0)
A = 0,
B = −3,
C = 0;
2
AC − B = −9 < 0.
$ %
!
& '
!
()
"
%
#*%+ z = f (x; y)
M
M
M
!
, ! z = f (x; y)
' G
! - ' %
& - - ' ' .
. ( '
' "
' / !% 0 ' . . !
' G -
( - ! z = f (x; y)% 1 '
G
!
z = f (x; y) G
!
"
G #
! $
z = f (x; y)
G
"
#
$
$
$
$ #
%
&' ( #
z = x2 − y 2 x2 + y 2 4
zy
) * +
= −2y ) $
"
2x = 0,
−2y = 0,
zx = 2x
P0 (0; 0) $
+$
# x2 + y 2 = 4 , $ #
z = x2 − y 2 #
$
$ x : z = x2 − (4 − x2) z = 2x2 − 4 % −2 x 2 -
# $
$
# x2 +y 2 = 4 #
$
$ $ z = 2x2 − 4
[−2; 2] +$
(−2; 2)
- z = 4x 4x = 0
x = 0. z|x=0 = −4 , z|x=−2 = 4 z|x=2 = 4 /
&
−4
z
z=f(x;y)
y
L
x
!!" # $ %
#&' y '
x '
!! # y ' &
x
( $' ' $ # ' # !!"
) * +#& %
,
- y x' !!" '
u . x' ' &
u x
' /0$ $
du
∂f
∂f dy
=
+
⇒
dx
∂x ∂y dx
∂f
∂f dy
+
= 0.
∂x ∂y dx
!!1
2 / !!" x' & 3
∂ϕ ∂ϕ dy
+
= 0.
∂x
∂y dx
( # & x y '
!!"
!!4
0& %
λ
!"#
$
%& ! '
∂f
∂ϕ
+λ
∂x
∂x
+
∂f
∂ϕ
+λ
∂y
∂y
dy
= 0,
dx
()
$"# $ !
* λ & $ ()
+ !+'
,- .
()'
∂f
∂ϕ
+λ
= 0.
∂y
∂y
$ x y !
!"
∂f
∂ϕ
+λ
= 0.
∂x
∂x
, .& ! $" $ ! $'
⎧
∂f
∂ϕ
⎪
⎪
⎪ ∂x + λ ∂x = 0,
⎨
∂ϕ
∂f
+λ
= 0,
⎪
⎪
∂y
∂y
⎪
⎩
ϕ(x; y) = 0.
(
, ( $$ $ ! ! -
! / &
! () $$" $
. ! 0 -
F (x; y; λ) = f (x; y) + λϕ(x; y)
((
x& y λ
, .& $ $ ! - ! !
1 ! 2 0 - !
+ +!" ! " ((& $ !"
. x& y λ& . ! (
+ . x y - + + λ ! /
& +! ! $ ( $$" $ !
& ! $ +
3- 4 . ! $ ! +
. . - .
5 $ $ ! "-
3xyz
λ = −
2a
⎧
3x
⎪
⎪
yz
1
−
(y
+
z)
= 0,
⎪
⎪
2a
⎪
⎪
⎪
⎪
⎨xz 1 − 3y (x + z) = 0,
2a
⎪
⎪
3z
⎪
⎪
(x
+
y)
= 0,
xy
1
−
⎪
⎪
2a
⎪
⎪
⎩
xy + xz + yz − a = 0.
x y z
⎧
3x
⎪
⎪
1 − (y + z) = 0,
⎪
⎪
2a
⎪
⎪
3y
⎨
1 − (x + z) = 0,
2a
⎪
3z
⎪
⎪
(x + y) = 0,
1
−
⎪
⎪
2a
⎪
⎩
xy + xz + yz − a = 0.
x = y !
y = z
x=y=z=
a
.
3
"
"# " x = 0
y = 0 z = 0
$ $"# "
"!% " &
" "
a
3
' (
!
"!%
) %
! "!%
&*
z = f (x; y)
P0 (x0; y0)
grad f (P0 )
! P1 (x1; y1) grad f (P1)
"#$ "# %
grad f (P1) P2 (x2; y2)
& ' (
# " " | grad f (Pn)|
$
)
" & $
'
*
! +
, $ -
$
./ /
− 6x − 3y.
z = x2 + xy + y2 −
0 1 ( $
$
! +
$1
⎧
⎪
⎨
⎪
⎩
∂z
∂z
= 2x + y − 6;
= x + 2y − 3
∂x
∂y
∂z
= 0,
2x + y − 6 = 0,
x = 3,
∂x
⇔
⇔
∂z
x
+
2y
−
3
=
0
y = 0.
=0
∂y
( $
2
· zyy
− (zxy
)
Δ = zxx
zxx
= 2;
zyy
= 2;
zxy
= 1 ⇒ Δ = 2 · 2 − 1 = 3 > 0.
! $
M(3; 0) $ ! 2 zxx
> 0+
!
31 4 M(3; 0) $
− 12x − 15y
z = y3 +3x2y −
zx
= 6xy − 12; zy = 3y 2 + 3x2 − 15
2
zx = 0,
x + y 2 = 5,
6xy − 12 = 0,
⇔
⇔
⇔
2
2
zy = 0
xy = 2
3y + 3x − 15 = 0
2
x + 2xy + y 2 = 9,
(x + y)2 = 9,
x + y = ±3,
⇔
⇔
⇔
2
2
x − y = ±1.
x − 2xy + y = 1
(x − y)2 = 1
M1 (1; 2);
M2 (2; 1);
M3 (−1; −2);
M4 (−2; −1).
2
Δ = zxx
zyy − (zxy
)
! "
zxx
= 6y; zyy
= 6y; zxy
= 6x ⇒
2
Δ = 6y6y − (6x) = 36(y 2 − x2 )
Δ|M1 = 36(4 − 1) > 0; Δ|M2 = 36(1 − 4) < 0
Δ|M3 = 36(4 − 1) > 0; Δ|M4 = 36(1 − 4) < 0.
# $ " M1 M3 $ ! " M2
M4 %
zxx
" M1 M3
zxx
|M1 = 6 · 2 > 0;
zxx
|M3 = 6 · (−2) < 0.
# $ M1 &
'
(
z = 2x2 + (y − 1)2
M3
%
)" % !
∂z
= 4x;
∂x
∂z
= 2(y − 1)
∂y
⎧
∂z
⎪
⎨
= 0,
x = 0,
4x = 0,
∂x
⇔
⇔
∂z
y = 1.
2(y
−
1)
=
0
⎪
⎩
=0
∂y
Δ = zxx zyy − (zxy )2
zxx
= 4,
zyy
= 2,
zxy
= 0 ⇒ Δ = 4 · 2 − 0 = 8 > 0.
M(0; 1)
Δzxx > 0
! " M(0; 1) #
$% $
z = 1 − x2 − y 2
x + y − 1 = 0
& ' ( ) ) z = 1 − x2 − y2
) * +,- L : x + y − 1 = 0
Oxy
z
N
z= 1-x 2-y
M
P
A
2
0
x+y-1=
B
y
x
. / 0
1 1
/ ' ) P 2 ; 2 0
1 0 A(1; 0) B(0; 1) '
2 A B
3 2 M
/ 2 N
. x + y − √1 = 0 y = 1 − x !
z = 1 − x2 − (1 − x)2 ⇔ z = 2x − 2x2 !/ x
2x − x2 0 ⇔ x ∈ [0; 1]
zx =
2 − 4x
√
2 2x − 2x2
z 12 =
1
zx = 0 ⇔ x0 =
2
1
= √ z(0) = 0 z(1) = 0
2
! " !
1
1
1
1
=√
x 0 = ⇒ y0 = 1 − x 0 = ⇒ z
2
2
2
2
x1 = 0 ⇒ y0 = 1 − x0 = 1 ⇒ z(0) = 0
x2 = 1 ⇒ y0 = 1 − x0 = 0 ⇒ z(1) = 0#
$%#& z = 6 − 3x − 4y
x2 + y2 = 1
' ( ) *"
F (x; y; λ) = 6 − 3x − 4y + λ(x2 + y 2 − 1).
!
⎧
3
⎪
⎧
⎧
⎪
x=
,
⎪
⎨
⎨ Fx = 0,
⎨ −3 + 2λx = 0,
2λ
2
F = 0, ⇔
−4 + 2λy = 0,
⇔
⇔
⎪ y= ,
⎩ y
⎩ 2
⎪
x + y 2 − 1 = 0.
Fλ = 0.
λ
⎪
⎩ 2
x + y 2 − 1 = 0.
⎧
⎧
3
3
⎪
⎪
⎪
x
=
,
x=
,
⎪
⎪
⎪
⎪
2λ
⎪
⎪
2λ
⎨
⎨
2
2
y= ,
⇔
⇔
y= ,
⎪
⎪
λ
λ
2
⎪
⎪
⎪
⎪
2
⎪
⎪
5
2
25 − 4λ
⎩
⎪
⎩
λ=± .
− 1 = 0.
= 0.
2
2
λ
4λ
⎧
⎧
3
3
⎪
⎪
x2 = − ,
x1 = ,
⎪
⎪
⎪
⎪
⎪
⎪
5
5
⎨
⎨
4
4
⇔
y1 = ,
y2 = − ,
⎪
⎪
5
5
⎪
⎪
⎪
⎪
⎪
⎪
5
⎩
⎩ λ = 5.
.
=
−
λ
1
2
2
2
⎧
3
⎪
x=
,
⎪
⎪
⎪
2λ
⎪
⎨
2
y= ,
⇔
λ
⎪
2
⎪
⎪
⎪
3
⎪
⎩
+
2λ
d2F = Fxx dx2 + 2Fxy dxdy + Fyy dy2
Fxx
= 2λ; Fyy
= 2λ; Fxy
= 0 ⇒ d2 F = 2λdx2 + 2λdy 2
5
3
4
λ1 = 2 x1 = 5 y1 = 5 d2F = 5(dx2 + dy2) > 0
λ2 = − 52 x2 = − 35 y2 = − 45 d2F = −5(dx2 + dy2 ) < 0
4
−4
= 1,
5
4
3
z = 6 − 3 −
−4 −
= 11.
5
5
z = 6 − 3
3
5
! " #z $ ! " #z $
z = 6 − 3x − 4y % x2 + y2 = 1
&' M − 35 ; − 45
M
3 4
;
5 5
.
()*
z = x2 − xy + y2 − 4y − x x 0 y 0 3x + 2y − 12 0
+ , ' - % . %
zx = 0,
⇔
zy = 0.
/
2x − y − 1 = 0,
⇔
−x + 2y − 4 = 0.
2 > 01 3 > 01 3 · 2 + 2 · 3 − 12 = 0
M(2; 3)
x = 2,
y = 3.
0 '
- . % z|M = 22 − 2 · 3 + 32 − 4 · 3 −
% 0 / x = 0
− 2 = −7 2 . %
zI = y 2 − 4y 2y − 12 0 y 0 -
,
0 ,
. % [0; 6]
- 3 (zI )y = 2y−4
(zI )y = 0 ⇔ 2y−4 = 0 ⇔ y = 2 -
. %
zI |y=2 = 22 − 4 · 2 = −4 -
. %
% '
zI |y=0 = 0 zI |y=6 = 62 −4·6 = 12
0 ,
x = 0 . % y = 6 : (zI ) = 12 , 4
y = 2 : (zI ) = −4
y = 0
y = 0 x = 4 : (zII ) = 12
1
1
x = : (zII ) = −
2
4
3x + 2y − 12 = 0
! " y = 6 − 3x
#
2
19x2
− 19x + 12
zIII =
4
x 0,
x 0,
x 0,
3x
⇔
⇔
y 0.
x 0.
0.
6−
2
x
!
"
zIII [0; 4] #!
"
x = 2 y = 3 : zIII =
= −7 ! $ x = 0 y = 6 : zIII = 12 %& '
$
( !) *
!
&)!
) "
z = 12 ! "
M1 (0; 6)
M2 (4; 0) z = −7 $ M3 (2; 3)
+, z = 12 z = −7
!
& - .$ - /
) "
z = x + xy + y − x − 2y
z = x4 + y4 − 2x2 + 4xy − 2y2
z = y2 − 2x2 − 2y + 1
2
- .
2
'
º
- 0
º
º
- /
& - - $ - -- ) ' )
1) "
- -
- --
z = 2x + y
z = x2 + y 2
x2 + y 2 = 5
2x + 3y = 6
& - -2$ - -3
"
!
!
z = xy + x + y
- -2
1 x 2, 2 y 3.
z = xy
x2 + y 2 1.
z = x2 + y 2 z = 2x + y
M
!" # $
" % & ' ( $!
)
*
+, ( $
-
. w = f (z)
z M
z ∈ M
w ! " w = f (z)
# $
M
% " % N
w " %
/ $ z - 0 w
1 , 0 2 w = f (z) 3 , $ (3 3 M z 3
N w
.. & " w = f (z) M
z w
4 0 $
+ ( z = f −1 (w)0 (!
32+ 3 N w 3 M z
2+ 3 $
z
w
n ∈ N w = z n
w = ρ(cos θ + i sin θ)
!
|z|n
z = r(cos ϕ+i sin ϕ)
ρ=r ,
θ = nϕ.
w = zn
z
( ($
'
n
" #
'
$ %&
(n − 1) arg z
k 2π
< ϕ < (k + 1) 2π
n
n
! w
% '
k = 0, 1, . . . , n − 1
(
'
'! !(
w=
√
n
z ) n&
M '
(% %
!
√
n
z
n
√
w = n z
n
,
x = 0 y = ϕ
( ** +'
$
w = ez
z = x + iy
'
!
z = 0 '&
(
ez = ex (cos y + i sin y)
&
-(
**.*
eiϕ = cos ϕ + i sin ϕ.
-
!
e2πki
2πi /!
= cos 2πk + i sin 2πk (k ∈ Z) !
ez+2πki = ez · e2πki = ez · 1 = ez .
/
w
!
1 '
z ' ! ln z
ew = z -0
% &
w = ln z
! !
z = 0
z % ' &
( !( &
2π
ln z = ln r + i(ϕ + 2πk), k ∈ Z.
ϕ = 0 r = 0
ln z k = 0
!" # # ln z ln z
k ln z $
ln z # z = 0
% & ' #& '
' ' w = za a = α + iβ ( za = ea ln z
)* + ln(−1).
, ( - ! + −1 = 1(cos π + i sin π) z = −1 |z| = r = 1 ϕ = π .
ln z ln(−1) = ln 1 + i(π + 2πk) = i(π + 2πk), k ∈ Z.
! # ln(−1)
sin z cos z
−iz
−iz
$ sin z = e −2ie / cos z = e +2 e .
0 1
' ##& 2
- eiz = cos z + i sin z
3 " ' 44
2π sin(−z) = − sin(z) cos(−z) = cos(z) sin2 z + cos2 z = 1 sin 2z =
= 2 sin z cos z cos 2z = cos2 z − sin2 z 5 sin z
cos z #
)* * cos i
, ( - .
iz
2
cos i =
iz
2
ei + e−i
e−1 + e1
=
≈ 1,54.
2
2
! # ( +
)* 4
w = f (z) z = x + iy w =
= u(x; y)+iv(x; y)
z0 = x0 + iy0 lim f (z)
z→z0
lim f (z) = x→x
lim u(x; y) + i x→x
lim v(x; y).
z→z0
0
y→y0
0
y→y0
w = f (z)
z0
z→z
lim f (z) = f (z0 ) f (z)
D
0
f (z) f (z)
f (z + Δz) − f (z)
.
Δz→0
Δz
$ f (z)
f (z) = lim
! " #
% z
& ' ($
% )
z$ w = f (z)
* ! f (z) = z2 − √z+3ez −
− 5 ln z + sin z + cos z + tg z
+ , -
1
1
5
.
f (z) = 2z − √ + 3ez − + cos z − sin z +
z
cos2 z
2 z
! ( w . u = u(x; y) %
. v = v(x; y)$ w = u + iv
( .
/ "# $%&'() * f (z) =
= u(x; y) + iv(x; y) z
u(x; y) v(x; y) (
( f (z) z +(
⎧
(
∂u
∂v
⎪
⎨
=
,
∂x
∂y
∂u
∂v
⎪
⎩
=− .
∂y
∂x
' /)
∂v
∂v
∂u
∂u
∂u
∂v
∂u
∂u
+i
=
−i
=
−i
=
+i .
∂x
∂x
∂y
∂y
∂x
∂y
∂y
∂x
f (z) =
f (z) = z̄
#$ %% !
!
z̄ = z − iy
z = x + iy
"
∂v
∂u
∂v
∂u
= 1;
= −1 ⇒
=
,
u(x; y) = x, v(x; y) = −y →
∂x
∂y
∂x
∂y
&
' #
⇒
' #
f (z) = z̄
'"
( f (z)
D
!
)
' # *
+
+
+ +
$
+
+
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y,
!
*
+ ! +
+ .
+, !
2
v(x; y)
!
2
"
x,
2
∂ u ∂ u
+
= 0.
∂x2 ∂y 2
12!
, $ ! $
∂ 2u
∂ 2v ∂ 2u
∂ 2v
=
/
=−
2
2
∂x
∂x∂y ∂y
∂x∂y
0 !
*
+ ! $ ! + ' #
%
2
∂ v ∂ v
+
= 0.
∂x2 ∂y 2
3 %, $ 4 , ' "
#,
6
$5
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0
! 7
, $ 2 ! "
, +
! ! +
2 ! ' #
+ ! * *,
4 ,
888 ! 5 2
= x3 − 3xy 2
ϕ(x; y) = x y ψ(x; y) =
3 2
ϕ(x; y)
ψ(x; y)
!
∂ϕ
= 3x2 y 2
"
# $%
# % &
∂x
∂ 2ϕ
∂ϕ
∂ 2ϕ
∂ 2ϕ ∂ 2ϕ
= 2x3 y
= 6xy 2
= 2x3
+ 2 = 6xy 2 + 2x3 = 0
2
2
∂x
∂y
∂y
∂x2
∂y
' ϕ(x; y) = x3 y 2 (
!
)
( "( ψ(x; y) *
∂ψ
∂ 2ψ
∂ψ
∂ 2ψ
∂ 2ψ ∂ 2ψ
= 3x2 − 3y 2 +
= −6xy +
= 6x+
= −6x
+
=
2
2
∂x
∂x
∂y
∂y
∂x2 ∂y 2
3
2
= 6x − 6x = 0 ( , ψ(x; y) = x − 3xy (
-
(
ψ(x; y) = x3 − 3xy 2 !
)
' "(
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(
/ #" //%
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a !
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3 (1
ez = 1 + z +
sin z = z −
z2 z3
+
+ ...
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3!
z3 z5
+
−...
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# /%
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n=0
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2
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z +
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|z| < 1
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n
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z||z − a| < R
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R=
)
1
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n→+∞
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2n
2 '
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¾
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n
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n=0
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"
$ %
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r < |z − a| < R 0 r +∞ 0 R +∞ % # (
#
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f (z) =
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Cn (z − a)n ,
n=−∞
) '') & % #
* !"!# $ %
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−1
Cn (z − a)n = C−1 (z − a)−1 + C−2 (z − a)−2 + . . .
+
n=−∞
+∞
$
&' Cn(z − a)n = C0 +
n=0
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' $
& !"!#
-
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lim f (z)
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a |f (z)|
z → a f (z) z → a
a
lim f (z)
z→a
f (z)
!
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! f (z) " # $
% &
' f (z) =
+∞
n=0
Cn (z − a)n
a (
! f (z) "
# $ %
& ' f (z) =
=
+∞
Cn (z − a)n
n=−k
a
! f (z)
" # $ %
& '
+∞
f (z) =
n=−∞
Cn (z − a)n
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# $ ( +
1
( &
!
z = a
f (z)
!
1
f (z)
n − 1 #
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-
-
! f (z) !
" z " "
' ( ( )
*
! + , )
( ( sin z( cos z !
z = 0
sin z
f (z) =
z
! " z
z2 z4
sin z
=1−
+
− ...
z
3!
5!
# $ % z = 0 $ & $
' $ ( $ !
cos z
f (z) =
)
z = 0
z
)
! " z
cos z
1
z
z3
= − +
− ...
z
z 2! 4!
#
$ % z = 0 *+,
( $ ! - ' z = 0 ,
1
z
=
!
.$ z = 0 &+
f (z)
cos z
z
/ !
0
cos z
z cos z + z sin z
=
cos z
cos2 z
z = 0
f (x) = e1/z
1
z = 0
1
1
1
% e1/z = 1 + +
+
+...
2
z 2!z
3!z 3
2 ' & ' ' ' +
' z = 0 ! f (z) = e1/z /
& $
z = arcsin(4x2 + y 2 )
4x2 − 36y − 3z 2 = 0
√
√
x x−x+ y
z =
y
∂z
dz
∂x
dx
z = xy y = ctg x
y y
ln y + x = ln x
z = √x + 2y
M(4; 2) N (5; 3) grad z(M)
! " z = x2 + xy +
2
y
+ +x−2y # $ m $ M
2
x 0 y ≥ 0 y 10+x
M0 (1; 1; 3/2) %
"
⇒
|4x2 + y 2 | 1 ⇒ −1 4x2 + y 2 1 ⇒
4x2 + y 2 ≥ −1,
2
2
4x + y 1,
,
z2
x2
−
⇒
9
12
Ox
4x2 − 36y − 3z 2 = 0 ⇔ y =
1
a = ;b = 1
2
y
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1
1/2
-1/2
x
-1
√
√
x x−x+ y
x3/2 − x
⇔z=
+ y −1/2 ⇒
y
y
√
3 1/2
x −1
3 x − 2 ∂z
x3/2 − x 1 − 3
∂z
= 2
=
=−
− y 2
⇒
∂x
y
2y
∂y
y2
2
3 1/2
x −1
∂ 2z
∂ 2z
3
2(x3/2 − x) 3 −5/2
4
= √
=
=
+ y
∂x2
y
y3
4
4 xy ∂y 2
√
3 1/2
x
−
1
2
−
3
∂ 2z
x
= −2
=
∂x∂y
y2
2y 2
z=
∂z
∂z
= y · xy−1 ;
= xy ln x ⇒
∂x
∂y
dz
∂z ∂z dy
1
=
⇒
=
+
= y · xy−1 + xy ln x − 2
dx
∂x ∂y dx
sin x
xctg · ln ctg x
= ctg x · xctg x−1 −
.
sin2 x
z = xy ; y = ctg x ⇒
1
y
y
ln y + x = ln x ⇔ + 1 = ⇒ y = − y = y
y
x
x
1
−1
x
y =
y
=y
1
1
−1 −y 2 =
x
x
x
2
1
y
2
−1 − 2 = y 1−
.
x
x
x
1
−1
x
= y
√
x + 2y M(4; 2) N (5; 3) ⇒
√
1
1
MN = (1; 1) ⇒ |MN | = 2 ⇒ cos α = √ ; cos β = √ ⇒
2
2
∂z
∂z
∂z
=
cos α +
cos β =
⇒
∂l M
∂x M
∂y M
1
1
1
2
√
√ + √
√
=
=
2 x + 2y 2 2 x + 2y 2 x = 4
y=2
z=
1 1
3
1 1
= √ √ +√ √ = ,
8
2 8 2
8 2
∂z ∂z
1
1
grad z(M) =
i
+
j = √ i + √ j.
∂x M
∂y M
2 8
8
y2
+ x − 2y ⇒
2
∂z
∂z
∂ 2z
∂ 2z
= 2x + y + 1;
= x + y − 2;
=
2;
= 1;
∂x
∂y
∂x2
∂y 2
⎧
∂z
⎪
⎨
= 0,
∂ 2z
x = −3,
2x + y + 1 = 0,
∂x
⇔
=1⇒
⇔
∂z
y
= 5.
x
+
y
−
2
=
0
⎪
∂x∂y
⎩
=0
∂y
2
∂ 2z ∂ 2z
∂ 2z
∂ 2z
Δ=
−
= 2 − 1 > 0; 2 = 2 > 0 ⇒
2
2
∂x ∂y
∂x∂y
∂x
x = −3; y = 5; z = −6,5.
x 10 + x x 0 y ≥ 0 !
" # $%& x = 0 y = 0
" # ABC #
z = x2 + xy +
y = x + 10 M(−3; 5)
z = −6,5
ABC
B(0;10)
K(-3,8;6,2)
M(-3;5)
D(0;2)
A(-10;0)
E(-0,5;0)
C(0;0)
! "
y2
dz
− 2y # y ∈ [0; 10]
= y − 2 = 0 ⇒ y = 2
• x = 0" z =
2
dy
$ m M
B(0; 10)
C(0; 0) D(0; 2)
dz
1
= 2x+1 = 0 ⇒ x = −
• y = 0" z = x2 +x # x ∈ [−10; 0]
dx
2
! m M %
1
%& ' A(−10; 0) E(− ; 0)
2
(x + 10)2
+ x − 2(x + 10) =
• y = x + 10" z = x2 + x(x + 10) +
2
dz
5 2
= x + 19x + 30(
= 5x + 19 = 0 ⇒ x = −3, 8 m M
2
dx
' K(−3, 8; 6, 2)
z(A) = z(−10; 0) = 90 z(B) = z(0; 10) = 30
z(C) = z(0; 0) = 0 z(D) = z(0; 2) = −2 z(E) = z(−0, 5; 0) = −0, 25
z(K) = z(−3, 8; 6, 2) = −6 z(M) = z(−3; 5) = −6, 5 %! )
!& z(A) = z(−10; 0) = M = 90 & z(M) = z(−3; 5) =
= m = −6, 5
*
z − z0 = fx (x0 ; y0 ) · (x − x0 ) + fy (x0 ; y0 )(y − y0 )
x0 = y0 = 1 z0 = 3/2 fx (x0 ; y0 ) = 2x0 + y0 + 1 = 4
fy (x0 ; y0 ) = x0 + y0 − 2 = 0
z − 3/2 = 4(x − 1)
8x − 2z − 5 = 0
z = ln(x2 − y 2 − 4)
4x − 2y2 − z2 = 0
x
z = arcsin y
dz
dx
z = u −1 v u = sin x v = √x
2
dy d y
dx
dx2
2y = x + ln x
!"
z =
M(3; 4)
# $ " z = x2 + y2 +
+y −2x m M 0 x 2 0 y 3
M0(1; 0; −1)
"% $
x2 + y 2
!" ! " # " $
% &
' & &
( ) &
*& & & ) " + )
) ! !,) $ (
!"
- & & +
, ) . !" f (x)
[a; b] ! , !", F (x) !
+
!" f (x)
F (x) = f (x).
/0123
4 & ,* & ! 5 ).
!" f (x) ! , !", F (x) " !
f (x)dx
dF (x) = F (x)dx = f (x) dx.
/0163
012 F (x) [a; b]
f (x)
f (x) [a; b]
%) ) F (x) [a; b]
# ! &
!" f (x) = cos 2x
!" F (x) = 12 sin 2x ! ! !
1
sin 2x = cos 2x
2
d
1
sin 2x = cos 2x dx.
2
% ! !"
, ) + ) 5 & ! .
!
"# $ % # &
'
() ( (
) * )
( (
+ ( , '
- #
. f (x) [a; b]
F (x)
F (x) + C C ! "
/ F (x) 0
f (x) F (x) + C 0 * (F (x) = f (x) ⇒ (F (x) + C) = f (x)) / '
[a; b]
f (x) ( '
( F (x) Φ(x) + % !& ,
Φ (x) = f (x),
F (x) = f (x).
1( ( '
() )
# )
,
(Φ(x) − F (x)) = 0.
% &
2 ( 3 * *
*
'
# * "#
( % &
Φ(x) − F (x) = C
Φ(x) = F (x) + C.
% &
1 % & C 0 4 (*
F (x) + C )( ) ()
'
2 (
cos 2x
1
x3
sin 2x + C, (
+ C. '
x2
2
3
( C * ( '
* ( ( - (
x1 ln x + C1 x > 0 ln(−x) + C2 x < 0
C1 C2
! x1 "
ln |x| + C #
C x > 0 x < 0
$
%&' F (x)
f (x) F (x) + C C
!
( #
$
#
f (x) dx )
f (x) dx = F (x) + C.
*%&+,
- $
,
f (x)
− ,
f (x) dx
−
.
/
$
−
cos 2x dx =
$
x2 dx =
1
sin 2x + C,
2
x3
+ C.
3
0# *
!, "
- 1# 2 "
1 2
( ! 3 #
# f (x) F (x) "
# #
!
! " !# $ % & "
' # #
$
$
sin x
dx −
x
2
e−x dx −
! ,
!.
(
) *#
$
+ # % ! & "
, !- % ! &
" !
f (x) dx,
k = 0.
. ! k
$
$
kf (x) dx = k
/
,0-
1 2 $! ! &
" ! $ % & " !/
$
$
(f (x) ± φ(x)) dx =
$
f (x) dx ±
φ(x) dx.
,3-
& ,0- ,3-
&&" ! 1 !
, 4 -
. $
! & " /
$
f (x) dx = f (x).
,5 + &&" $
/
$
d
f (x) dx = f (x) dx.
,6 !
$
dx
= d arcsin x,
1 − x2
dx
= d ln x.
x
√
!
" # ! $
! $ !
%
! & $
! ' '
( ! $ # sinx x dx e−x dx
' ) * $
$ sinx x e−x + ,
2
2
- !$
. $ !
/ 0 !. $
* ) *
0 $ $ 0 *
* $
" '
$ *
* ! *
!*
1 !
*
$ * n+1
2 n+1
$
xn dx =
1.
d
! $
x
x
=
+ C, (n = −1),
n+1 $ n+1
n=0
dx = x + C,
$
dx
= d(ln |x|) = ln |x| + C,
$ x
$
3. sin x dx = d(− cos x) = − cos x + C,
$
2.
$
4.
$
cos x dx = d(sin x) = sin x + C,
$
dx
=
d(tg x) = tg x + C,
5.
2
$ cos x $
dx
6.
= d(− ctg x) = − ctg x + C,
2
sin
x
$
$
x
1
x
1
dx
arctg
= arctg + C,
=
d
7.
2 + x2
a
a
a
a
a
$
$
dx
x
x
√
8.
= arcsin + C,
= d arcsin
a
a2 − x$2
$
$ a
x
x
a
a
x
9.
a dx = d
=
+ C,
ex dx = ex + C,
ln a
ln a
$
√
dx
√
= ln |x + x2 ± a2 | + C,
10.
x2 ± a2
$
$
x − a
a + x
1
1
dx
dx
+ C,
+ C.
ln
ln
=
=
11.
x2 − a2
2a x + a
a2 − x2
2a a − x
$
!
" # # $ % #
& # % & &" % ' %
& ( & % ) # #&
& % &
*+ ,
x
x
*+ +
! " n < 0# $ % & !' !!
( C )
( )
% - %
# -
) # # #
%
. # ) "
% # &
$
$
! $" #
f (x)dx =
d
f (φ(t))φ (t)dt,
f (x)dx = f (x)dx,
$
d
f (φ(t))φ(t)dt = f (φ(t))d(φ(t)) = |x = φ(t)| = f (x)dx.
$ " % "
&% '( x = φ(t) t : t = ψ(x)
% Φ(t) :
$
$
f (x)dx =
f (φ(t) · φ (t))dt = Φ(t) + C = Φ(ψ(x)) + C.
#
f (x)dx
x φ(t) dx φ(t)dt
)
!"
# "
ϕ(x) = t
#
$%
I = x2ex dx.
* ( # + % % ,
- x3 x2. . % x
( ' +/,
' x3 = t, ,
% " 3x2dx = dt. .
x2 01 dt.
2 % -3 3,
- '#
3
1
I = x3 = t ⇒ 3x2 dx = dt =
3
4 $%
$
1
1 3
et dt = et + C = ex + C.
3
3
I =
#
sin kx dx.
dt
k
$
1
1
1
sin tdt = − cos t + C = − cos kx + C .
I=
k
k
k
kx = t =⇒ dx =
$
I=
√
t=x+
√
dx
.
± a2
x2
x2 ± a2 ⇒
√ 2
x ± a2 + x
x
= √
dx ⇒
⇒ dt = 1 + √
2
2
x ±a
x2 ± a2
dt
dx
dt
dx
√
=
⇒√
= .
⇒√
t
x2 ± a2
x + x2 ± a2
x2 ± a2
$
I=
√
dt
= ln |t| + C = ln |x + x2 ± a2 | + C.
t
! " # $
% & $ ' " #
'( #" " ) *+
$ $
I = # (x − 3)2dx
$
I=
$
2
(x − 6x + 9)dx =
2
x dx − 6
$
$
xdx + 9
dx =
x3
x2
x3
+ C1 ) − (6 + C2 ) + (9x + C3 ) =
− 3x2 + 9x + C.
3
2
3
, -
*( ' ( . ( C1 − C2 + C3
' . . .- / & ' C
=(
+
( & $ '( $ (
' ' ' $
0 I =
$
x2 − 3
√ dx.
x
$
3
2
(x − 3x
I=
− 21
3
1
√
x− 2 +1
2 √
x 2 +1
)dx = 3
−3 1
+ C = x2 x − 6 x + C.
5
+
1
−
+
1
2
2
I =
$
I=
sin2 x + cos2 x
dx =
sin2 x cos2 x
$
$
dx
sin2 x cos2 x
dx
+
cos2 x
$
dx
= tg x − ctg x + C.
sin2 x
$
dx
=2
sin2 2x
I=4
$
$
d(2x)
= −2 ctg 2x + C.
sin2 2x
#
I = tg5 x dx.
$
1
dx
− 1 dx = tg 3 x 2 −
I = tg x tg xdx = tg x
cos2 x
cos x
$
$
$
$
tg 4 x
− tg 3 xdx.
− tg 3 xdx = tg 3 xd tg x − tg 3 xdx =
4
n = 5 n = 3
3
$
2
3
$
1
tg4 x
− tg x
−
1
dx =
4
cos2 x
$
$
tg4 x
tg4 x tg2 x
=
− tg xd tg x + tg xdx =
−
− ln | cos x| + C.
4
4
2
I=
!
1
(sin(n − m)x + sin(n + m)x) ,
2
1
sin nx sin mx = (cos(n − m)x − cos(n + m)x) ,
2
1
cos nx cos mx = (cos(n − m)x + cos(n + m)x) .
2
sin nx cos mx =
"
#$% &'(
$
I=
#
sin 2x cos 3x dx.
1
1
1
(− sin x + sin 5x)dx = cos x −
cos 5x + C.
2
2
10
I=
$
I=
dx
.
x2 − a2
1
1
=
x2 − a2
2a
1
1
−
x−a x+a
.
!"
!# $ % !"
$
$
d(x − a)
d(x + a)
1
−
=
I=
2a
x−a
x+a
x − a
1
1
+ C.
=
(ln |x − a| − ln |x + a|) =
ln
2a
2a x + a
& ' ( ) * ++
, d(uv) = udv + vdu- ) % )
" !
$
$
udv = uv − vdu.
. +/
0 !" . +/ ( 1 !" ) !
, ! )
2/ () 1"3 4
"# ) ! "# 1"3
$
$
$
Pn (x)eαx dx
Pn (x) sin αx dx
Pn (x) cos αx dx.
$
Pn (x)eαx dx =⇒
eαx
dv = e dx = d
α
Pn (x)dx,
αx
u = Pn (x), du =
$
Pn (x) sin αx dx =⇒
, v=
eαx
.
α
cos αx
cos αx
, v=−
.
u = Pn (x), du = Pn (x)dx, dv = sin αx dx = d −
α
α
$
Pn (x) cos αx dx =⇒
sin αx
sin αx
, v=
.
u = Pn (x), du = Pn (x)dx, dv = cos αx dx = d
α
α
!
" " "
$
Pn (x) ln x dx
$
Pn (x) arcsin αx dx
$
Pn (x) arctg αx dx.
$
Pn (x) ln x dx =⇒
dx
, dv = Pn (x)dx, v =
u = ln x, du =
x
$
Pn (x) arcsin αx dx =⇒
u = arcsin αx, du =
$
αdx
1 − (αx)2
$
Pn (x) dx,
$
, dv = Pn (x)dx, v =
Pn (x) dx,
Pn (x) arctg αx dx =⇒
u = arctg αx, du =
αdx
, dv = Pn (x)dx, v =
1 + (αx)2
$
#
$%&'( I = # x2 ln x dx.
Pn (x) dx.
u = ln x
du = dx
x =
I =
3
3
x
x
2
dv = x dx = d 3 v = 3
$
x3
1
x3
x3
=
ln x −
x2 dx =
ln x −
+ C.
3
3
3
9
#
I = x arctg x dx.
$
u = arctg x
1
du = x2dx+1 x2
x2 dx
arctg
x
−
=
=
I =
2
2
x
x
dv = xdx = d 2 v = 2
2
2
x2 + 1
$ 2
x2
1
x +1−1
x2
=
arctg x −
dx =
arctg x−
2
2
2
x +1
2
$
$
1
1
dx
x2
1
1
−
dx +
=
arctg x − x + arctg x + C.
2
2
2
x +1
2
2
2
# 2
I = x sin x dx.
u = x2
du = 2xdx
I =
dv = sin xdx = d(− cos x) v = − cos x
$
u=x
+ 2 x cos xdx =
dv = cos xdx = d(sin x)
= −x2 cos x + 2x sin x + 2 cos x + C.
I=
#
= −x2 cos x+
du = dx
=
v = sin x
xe3x dx.
$
u=x
du = dx 1 3x 1
e3x dx =
I =
−
1 3x = xe
3x
dv = e dx v = 3 e
3
3
1
1
= xe3x − e3x + C.
3
9
! """
# $%& & & ' (
$ )
* ! &
I = # eax cos nx dx.
u = eax
du = aeax dx
=
I =
1
dv = cos nx dx v = n sin nx
$
1
a
= eax sin nx −
eax sin nx dx =
n
n
u = eax
du = aeax dx 1 ax
=
= e sin nx−
dv = sin nx dx v = − n1 cos nx n
$
1
a
a
− eax cos nx +
eax cos nx dx =
−
n
n
n
a ax
a2
1 ax
= e sin nx + 2 e cos nx − 2 I.
n
n
n
I
$
I=
eax cos nx dx =
eax (a cos nx + n sin nx)
+ C.
a2 + n2
C !
"# $ "
√
%&'( I = # x2 − a2 dx.
√
$
√
u = x2 − a2 du = √ xdx
x2 dx
x2 −a2 = x x2 − a2 −
√
=
I =
dv = dx
v=x
x2 − a2
$
$
√
√
√
x2 − a2 + a2
√
= x x2 − a2 −
dx = x x2 − a2 −
x2 − a2 dx−
x2 − a2
$
√
√
dx
= x2 x2 − a2 − I − a2 ln |x + x2 − a2 | + C1 .
− a2 √
2
2
x −a
)
I=x
2
√
x2
−
a2
− I − a2 ln |x +
√
x2 − a2 | + C1 ,
√
1 √
a2
I = x x2 − a2 − ln |x + x2 − a2 | + C.
2
2
!
"
f (x) = cos12 2x .
# $ % F (x) = 12 tg 2x + C.
& "% f (x) = F (x) = 12 tg 2x = 21 · 2 cos12 2x = cos12 2x .
' f (x) = x4.
x5
+ C.
5
x5
1
+ C = · 5x4 + 0 = x4 .
f (x) = F (x) =
5
5
# $ %
& "%
F (x) =
f (x) = x2 .
# $ % F (x) = 2 ln | − x| + C.
& "% f (x) = F (x) = (2 ln | − x| + C) = 2(− x1 )(−1) + 0 = x2 .
'
( )
!
& *
(! + !
" ) ! ,
" "
# " " -
(!. / "
0 ! ! 1 !
%
I = # sin3 x cos x dx.
# $ %
2/!) / d(sin x) = cos xdx,
$
I=
sin3 xd sin x =
sin4 x
+ C.
4
$
I=
$ √
arctg x
dx.
1 + x2
I =
(arctg x)1/2 d arctg x =
d(arctg x) =
(arctg x)1/2+1
1/2 + 1
dx
,
1 + x2
2
= (arctg x)3/2 + C.
3
! " # $ %
& I1 =
#
tg x dx, I2 =
#
ctg x dx.
$
$
d cos x
sin x dx
=−
= − ln | cos x| + C.
I1 =
cos x
cos x
$
$
d sin x
cos x dx
=
= ln | sin x| + C.
I2 =
sin x
sin x
$
dx
.
' I =
x ln x
I=
$
dx
=
x ln x
$
dx/x
=
ln x
$
d ln |x|
= ln | ln |x|| + C.
ln x
!
#
( I = sin 5x dx.
!
# $
) * %
+ * $ +
15 :
$
1
1
sin 5x d5x = − cos 5x + C.
I=
5
5
#
, I = ex cos ex dx.
-* ** d(ex ) = ex dx,
$
I = cos ex dex = sin ex + C.
!"#$ I =
$
% &
dx
.
x2 cos2 x1
$
I=−
!"## I =
% &
I=
1
3
!"#'
% &
1
I=
4
$
!"#!
$
d x1
1
= tg + C.
x
cos2 x1
dx
.
sin2 (3x − 5)
$
d(3x − 5)
1
= − ctg(3x − 5) + C.
3
sin2 (3x − 5)
$
3
x dx
I=
.
5 − x8
√
5 + x4
dx4
1
√
= √ ln √
+ C.
( 5)2 − (x4 )2
8 5 5 − x4
#
2
I = x3x dx.
% &
1
I=
2
$
2
3x dx2 =
2
3x
+ C.
2 ln 3
%
( ) *)
&
$
$
$
(f (x) ± ϕ(x)) dx = f (x) dx ± ϕ(x) dx,
$
$
kf (x) dx = k f (x) dx.
I=
$
x4 − 10x2 + 5
dx.
x2
$
$
$
$
5
I=
x2 − 10 + 2 dx = x2 dx − 10 dx + 5 x−2 dx =
x
5
x3
− 10x − + C.
3
x
# √
2
dx.
I=
2x +
x
=
√ $ 1
√ $
√
1
I = 2 x 2 dx + 2 x− 2 dx = 2
#
I = cos 2x cos 5x dx.
√
2 √
x x + 2 x + C.
3
$
$
1
1
(cos 7x + cos 3x) dx =
cos 7xd(7x)+
I=
2
14
$
1
1
1
+
cos 3x d(3x) =
sin 7x + sin 3x + C.
6
14
6
! "# " " $ # "#
# "%& '"# "
$
√
3
( I = 9x2 x3 + 10 dx.
3
1
+1
x + 10 = t
#
3
= 3 t 13 dt = 3 t
I = 2
+C =
1
3x dx = dt
+1
3
√
√
3
= 94 t t + C = 94 (x3 + 10)3 x3 + 10 + C.
$
4xdx
√
) I =
.
5
8 − x2
− 15 +1
8 − x2 = t
#
= −2 t− 15 dt = −2 t
+C =
I =
−2x dx = dt
−1 + 1
4
4
= − 25 t 5 + C = − 52 (8 − x2 ) 5 + C.
5
2 cos x dx
.
4 + sin x
4 + sin x = t
I =
cos x dx = dt
I =
= 2 ln(4 + sin x) + C.
$
$
dt
= 2
= 2 ln |t| + C =
t
4 + sin x > 0
I=
$
arcsin x
dx.
1 − x2
arcsin x = t
√
#√
=
I = √ dx
t dt = 23 t t + C =
=
dt
1−x2
√
= 32 arcsin x arcsin x + C.
! "#
√
1 − x2 $ "
#
$ √
$ √
dx
I=
arcsin x √
=
arcsin x d(arcsin x) =
1 − x2
√
2
= arcsin x arcsin x + C.
3
%
&
#
#
I = ex (ex + 2)2 dx.
! "# $ "
'
#
(ex + 2)3
+ C.
I = (ex + 2)2 d(ex + 2) =
3
$ x
e + sin x
dx.
I=
ex − cos x
$
d(ex − cos x)
= ln |ex − cos x| + C.
I=
ex − cos x
$
x+5
√
dx.
I =
x2 + 3
$
$
dx
d(x2 + 3)
1
x dx
√
+5 √
=
+
I = √
2
x2 + 3
x2 + 3
x2 + 3
$
√
√
dx
= x2 + 3 + ln |x + x2 + 3| + C.
2
(x + 3)
I=
$
x4 dx
.
1 + x10
$
dx5
1
1
= arctg x5 + C.
5
2
5
1 + (x )
5
#
I = xeax dx.
$
u=x
du = dx xeax 1
−
eax dx =
I=
1 ax =
ax
dv = e dx v = a e
a
a
xeax eax
− 2 + C.
=
a
a
#
I = xn ln x dx (n = −1).
u = ln x
#
xn+1
du = dx
x
ln x+ xn−1 dx =
=
I =
n+1
x
n
dv = x dx v = n+1
n
+
1
1
xn+1
ln x −
+ C.
=
n+1
n+1
! " n = −1, #$ !
$
$
1
dx
= ln x d(ln x) = ln2 x + C.
I = ln x
x
2
# ax
% I = e sin nx dx.
I=
u = eax ,
1
du = aeax dx
= − eax cos nx+
I=
dv = sin nx dx, v = − n1 cos nx
n
$
u = eax
a
du = aeax dx
=
eax cos nx dx =
+
dv = cos nx dx v = n1 sin nx
n
$
a 1 ax
a
1
e sin nx −
eax sin nx dx =
= − eax cos nx +
n
n n
n
a
a2
1 ax
= − e cos nx + 2 − 2 I. ⇒
n $
n
n
eax (a sin nx − n cos nx)
ax
+ C.
⇒ I = e sin nx dx =
a2 + n2
# & !'
eax cos nx dx.
!
( #
f (x) = 6x2.
f (x) = tg 5x.
I = # √x dx 4 .
2−x
! I = # sincosx2xdx .
2
x dx
√
.
I = # cos
sin x
I = # (1 − 7x)5x dx.
I = # etg x cosdx2 x .
3
#
3
x
" I = arcsin
dx.
1 − x2
# x +2x
# I = e (x + 1) dx.
$ I = # √xx2dx− 6 .
2
I = # e
√
x
dx
√ .
x
I = # √9 − exex dx.
I = # (√x − 1)2 dx.
5
! I = # (x −x 1) dx.
I = # (sin 5x cos x) dx.
3
% &'
√
I = # x 2x2 + 7 dx.
+ 5) dx
.
I = # x(2x
2 + 5x − 13
I =
# 2 + ln x
.
x
# sin 2x dx
.
" I =
7 + cos2 x
# x + x3
dx.
# I =
x4 + 5
#
% I = x sin 2x dx.
#
& I = arctg x dx.
#
' I = e2t cos 3t dt.
! I =
(
#
sin 3x dx
√
.
5 + cos 3x
#√
dx
I = 3 tg 2x 2 .
cos 2x
$
) * + * , -
*
.
*
, * $ /*
A
,
I.
x−a
A
II.
(n = 2, 3, ...),
(x − a)n
Mx + N
III. 2
(D = p2 − 4q < 0),
x + px + q
Mx + N
IV. 2
(D = p2 − 4q < 0, n = 2, 3...).
(x + px + q)n
!
$
d(x − a)
Adx
=A
= A ln |x − a| + C.
x−a
x−a
$
$
(x − a)−n+1
Adx
+C =
= A (x − a)−n d(x − a) = A
II.
n
(x − a)
−n + 1
A
=
+ C.
(1 − n)(x − a)n−1
$
I.
"
# $ ax2 + bx + c #
"%
1
b
(ax2 + bx + c) = ax + = t.
2
2
1 2
(x + pq + q) = t dx = dt
Mx + N
dx = 2 p
III.
x=t−
x+ 2 =t
x2 + px + q
$
$
p
Mt + (N − Mp
M(t − 2 ) + N
)
2
dt =
dt.
=
p 2
p
P2
2
(t − 2 ) + p(t − 2 ) + q
t + (q − 4 )
&'' (
$
) % *
q−
p
2
=
p2
= a2 > 0,
4
$
$
dt
Mx + N
tdt
Mp
dx
=
M
+
N
−
=
x2 + px + q
t2 + a2
2
t2 + a2
t
Mp 1
M
ln(t2 + a2 ) + N −
arctg + C.
=
2
2
a
a
$
+ t a # *
$
=
Mx + N
dx =
x2 + px + q
N − Mp
x + p2
M
2
ln(x2 + px + q) +
arctg
+ C.
2
2
2
q − p4
q − p4
&''!(
! ! 2x + p. "
2x+p Mx+N. # 2x+p
N/2 N − Mp/2.
$
(2x + p)
Mp
M
+N −
= Mx + N.
2
2
%
Mx + N
x2 + px + q
(2x + p) M2 + N −
x2 + px + q
Mp
c
&
N − Mp
M 2x + p
2
+
.
2 x2 + px + q x2 + px + q
' #
( ) !
p 2
p2
+q− .
x2 + px + 1 = x +
2
4
*
+ ,,
+ - 4q − p2 > 0.
..!,
4q − p2 < 0,
ax2 + bx + c,
a
x2 + px + q
..!,
$
I=
x+1
dx
x2 + 4x + 8
t t = x + 2 x = t − 2 dx = dt
$
$
$
t−2+1
t−1
x+1
dx
=
dt
=
dt =
I=
x2 + 4x + 8
(t − 2)2 + 4(t − 2) + 8
t2 + 4
$
$
tdt
dt
1
t
1
=
−
= ln(t2 + 4) − arctg + C =
t2 + 4
t2 + 4
2
2
2
1
x+2
1
= ln(x2 + 4x + 8) − arctg
+ C.
2
2
2
! !
" " 2x + 4 #
$ %! ! & $ !' ($
x+1
1 2x + 4 − 2
=
.
x2 + 4x + 8
2 x2 + 4x + 8
' ! "
' " '
1 2x + 4 − 2
1 2x + 4
1
=
−
.
2 x2 + 4x + 8
2 x2 + 4x + 8 (x + 2)2 + 4
)
$
d(x2 + 4x + 8)
d(x + 2)
−
=
2
x + 4x + 8
(x + 2)2 + 22
x+2
1
1
+ C.
= ln(x2 + 4x + 8) − ln arctg
2
2
2
I=
1
2
$
*+ ! *+
! ***
$
Mx + N
dx =
(x2 + px + q)n
$
$
dt
Mp
tdt
+
N
−
.
=M
(t2 + a2 )n
2
(t2 + a2 )n
$
'
tdt
1
=
(t2 + a2 )n
2
$
,(( -.
,(( -. '! "!"
(t2 + a2 )−n d(t2 + a2 ) =
1
+ C.
2(1 − n)(t2 + a2 )n−1
$
In =
=
1
a2
#
In =
#
#
dt
(t2 +a2 )n
$ 2
(t + a2 ) − t2
dt
1
=
dt =
2
2
n
2
(t + a )
a
(t2 + a2 )n
$
$
dt
t2 dt
.
−
(t2 + a2 )n−1
(t2 + a2 )n
dt
(t2 +a2 )n−1
In =
t2 dt
(t2 +a2 )n
= In−1
1
a2
In−1 −
$
t2 dt
2
(t + a2 )n
.
u=t
du = dt
1
dv = 2 tdt2 n v =
(t +a )
2(1−n)(t2 +a2 )n−1
$
t2 dt
t
1
In−1 .
=
−
2
(t + a2 )n
2(1 − n)(t2 + a2 )n−1 2(1 − n)
! "
2n − 3
t
In−1 +
2n − 2
2(n − 1)(t2 + a2 )n−1
#
$ % &
% ' (
! )* & n &
! )* & n − 1
+ # , n − 1 & & ,
-
. I3 = # (t2 +dt 1)3
In =
/
, a
1
I3 = 2
1
I2 =
$
1
a2
0
= 1, n = 3
2·3−3
t
I2 +
2·3−2
2(3 − 1)(t2 + 1)2
.
#
3
t
= I2 +
.
4
4(t2 + 1)2
1
dt
2·2−3
t
t
I1 +
= I1 +
.
=
(t2 + 1)2
2·2−2
2(2 − 1)(t2 + 1)
2
2(t2 + 1)
$
I1 =
I2 =
dt
= arctg t + C,
t2 + 1
t
1
arctg t +
+C
2
2(t2 + 1)
1
t
t
arctg t +
+
+C =
2
2(t2 + 1)
4(t2 + 1)2
3
3t
t
+ arctg t + C.
+
=
2
2
2
4(t + 1)
8(t + 1) 8
3
I3 =
4
#
R(x)dx R(x)
# ! " #$ % &
R(x)dx
" #$
$
I=
2x2 + 5x − 8
dx.
(x − 1)3 (x + 2)2
D
A
B
C
E
2x2 + 5x − 8
+D
.
=
+
+
+
(x − 1)3 (x + 2)2
(x − 1)3 (x − 1)2 x − 1
(x + 2)2 x + 2
! "
2x2 + 5x − 8 = A(x + 2)2 + B(x − 1)(x + 2)2 + C(x − 1)2 (x + 2)2 +
+ D(x − 1)3 + E(x − 1)3 (x + 2).
#$%%& ' A, B, C, D, E ! (
! ') ! * ') $%%&
⎧
x = 1 ⇒ −1 = 9A ⇒ A = − 19 ,
⎪
⎪
⎪
10
⎪
⎪
⎨ x = −2 ⇒ −10 = −27D ⇒ D = 27 ,
x = 2 ⇒ 10 = 16A + 16C + D + 4E ⇒ 4B + 4C + E = 77
,
27
⎪
⎪
4
⎪
⇒
C
+
E
=
0,
x
⎪
⎪
⎩
⇒ −8 = 4A − 4B + 4C − D − 2E ⇒ 2B − 2C + E =
B, C, E
⎧
⎪
⎨C + E = 0,
2B − 2C + E =
⎪
⎩4B + 4C + E =
!
97
.
27
97
,
27
77
,
27
"
B=
13
13
29
, C=− , E= .
27
27
27
# $ !
$
$
2x2 + 5x − 8
−1/9
29/27
−13/27
+
dx
=
(
+
+
(x − 1)3 (x + 2)2
(x − 1)3 (x − 1)2
x−1
10/27
1
1
13/27
29 1
+
)dx =
−
+
−
(x + 2)2
x+2
18 (x − 1)2 27 x − 1
10 1
13
13
ln |x − 1| −
+
ln |x + 2| + C =
−
27
27 x + 2 27
13 x + 2
26x2 + 5x − 34
+
ln
+ C.
=−
18(x − 1)2 (x + 2) 27 x − 1
%%%
$
I=
x4 + 5x3 − 6x + 5
dx.
x3 + 2x2 − 1
&
" ' $! ( ) !
* $"
x4 + 5x3 − 6x + 5
−6x2 − 5x + 8
= x+3+ 3
.
3
2
x + 2x − 1
x + 2x2 − 1
x3 + 2x2 − 1
x = −1
x + 1
x3 + 2x2 − 1 = (x + 1)(x2 + x − 1).
! "# x2 + x − 1
$
%
&
−6x2 − 5x + 8
A
Bx + C
=
+
.
(x + 1)(x2 + x − 1)
x + 1 x2 + x − 1
' &
−6x2 − 5x + 8 = A(x2 + x − 1) + (Bx + C)(x + 1) =
= (A + B)x2 + (A + B + C)x − A + C.
!(( % A, B, C
)
# " # (( %
⎧
⎪
⎨ x = −1 =⇒ 7 = −A =⇒ A = −7,
x2 ⇒ −6 = A + B =⇒ B = 1,
⎪
⎩ ⇒ 8 = −A + C =⇒ C = 1.
*
−6x2 − 5x + 8
−7
x+1
=
+
.
(x + 1)(x2 + x − 1)
x + 1 x2 + x − 1
&
)
x+1
1 2x + 1 + 1
1 2x + 1
1
1
= · 2
=
+
.
x2 + x − 1
2 x +x−1
2 x2 + x − 1 2 (x + 12 )2 − 54
1
2x + 1
1
1
7
+ ·
+
I=
dx =
x+3−
x + 1 2 x2 + x − 1 2 (x + 12 )2 − 54
$
$
$
$
d(x + 12 )
d(x2 + x − 1) 1
1
+
=
= x dx − 3
dx +
2
x2 + x − 1
2
(x + 12 )2 − 54
√
1
1 x + 12 − 25
x2
√ + C.
+ 3x − 7 ln |x + 1| + ln |x2 + x − 1| + ln
=
2
2
2 x + 1 + 5
$
2
$
I=
4
2
3
x + 5x − 7x2 + 5
dx.
x3 − x2 + 5x − 5
!
x4 + 5x3 − 7x2 + 5
6x2 + 25x − 35
=
x
+
6
−
.
x3 − x2 + 5x − 5
x3 − x2 + 5x − 5
" "
3
x − x2 + 5x − 5 = x2 (x − 1) + 5(x − 1) = (x − 1)(x2 + 5).
" # !
A
Bx + C
6x2 + 25x − 35
=
+ 2
.
x3 − x2 + 5x − 5
x−1
x +5
$ !
#
6x2 + 25x − 35 = A(x2 + 5) + (Bx + C)(x − 1).
%&''( ) A, B, C ! )*
! +)* &''(
⎧
2
⎪
⎨ x = 1 ⇒ −4 = 6A =⇒ A = − 3 ,
2
x ⇒ 6 = A + B =⇒ B = 6 − A = 6 +
⎪
⎩ ⇒ −35 = 5A − C =⇒ C = 95 .
3
2
3
=
20
,
3
6x2 + 25x − 35
2
1
20x/3 + 95/3
=− ·
+
.
3
2
x − x + 5x − 5
3 x−1
x2 + 5
$
I=
x2
2
x2
=
2
=
1
20x/3 + 95/3
2
+
=
x+6+ − ·
3 x−1
x2 + 5
$
$
2
20
x dx
95
dx
+ 6x − ln |x − 1| +
+
=
2
2
3
3
x +5
3
x +5
x
1 (x2 + 5)10
95
+ 6x + ln
+ √ arctg √ + C.
3
(x − 1)2
3 5
5
! "
#
$ #
$
.
%%#& I = x dx
−5
' (
$
I=
d(x − 5)
= ln |x − 5| + C.
x−5
%%#)
' ($
I=
(x + 2)−4 d(x + 2) =
I=
x+3
dx.
x2 + 4x + 29
' (
$
dx
.
(x + 2)4
1
(x + 2)−4+1
+C =−
+ C.
−4 + 1
3(x + 2)3
%%#*
$
I =
$
1
t+1
dt =
I = | (x2 + 4x + 29) = x + 2 = t, dx = dt| =
2
t2 + 25
$
$
dt
1
t
1
t dt
+
= ln(t2 + 25) + arctg + C =
=
t2 + 25
t2 + 25
2
5
5
x+2
1
1
+ C.
= ln(x2 + 4x + 29) + arctg
2
5
5
I4 =
$
dx
.
(x2 + 1)4
$
dx
1 2n − 3
x
I
In =
=
+
;
n−1
(x2 + a2 )n
a2 2n − 2
2(n − 1)(x2 + a2 )n−1
5
x
;
I4 = I3 +
6
6(x2 + 1)3
3
x
;
I3 = I2 +
4
4(x2 + 1)2
1
x
.
I2 = I1 +
2
2(x2 + 1)
$
dx
= arctg x + C
I1 =
2
x +1
1
2
arctg x +
;
3
2(x2 + 1)
1
1
3 2
arctg x +
+
=
I3 =
2
2
4 3
2(x + 1)
4(x + 1)2
1
3
1
= arctg x +
+
;
2
8(x2 + 1) 4(x2 + 1)2
3
1
5 1
1
I4 =
arctg x +
+
=
+
4 2
8(x2 + 1) 4(x2 + 1)2
6(x2 + 1)3
5
15
5
1
= arctg x +
+
+
+ C.
2
2
2
2
8
32(x + 1) 16(x + 1)
6(x + 1)3
I2 =
I =
$
x4 − 3x3 − 5x2 + 30x − 22
dx.
x3 − x2 − 8x + 12
!
" # # $
x4 − 3x3 − 5x2 + 30x − 22
x2 + 2x + 2
=x−2+ 3
.
3
2
x − x − 8x + 12
x − x2 − 8x + 12
$
I=
=
x2 + 2x + 2
3
x − x2 − 8x + 12
x−2+
x2
− 2x +
2
$
x3
dx =
x2 + 2x + 2
dx.
− x2 − 8x + 12
x3 − x2 − 8x + 12 = (x − 2)2(x + 3)
x3
x2 + 2x + 2
x2 + 2x + 2
A
B
C
=
=
+
.
+
2
− x − 8x + 12
(x − 2)2 (x + 3)
x − 2 (x − 2)2 x + 3
x2 + 2x + 2 = A(x − 2)(x + 3) + B(x + 3) + C(x − 2)2 .
!"" A B C
# $ # "" %
I=
⎧
⎪
⎨ x = 2 ⇒ 10 = 5B ⇒ B = 2,
x = −3 ⇒ 5 = 25C ⇒ C = 15 ,
⎪
⎩ x2 ⇒ 1 = A + C ⇒ A = 4 .
5
4
2
1
x2
− 2x + ln |x − 2| −
+ ln |x + 3| + C.
2
5
x−2 5
&&'( I =
$
x2 − 5x + 9
dx.
(x − 1)2 (x2 + 2x + 2)
) % * +
) $ %
'
x2 − 5x + 9
A
=
+
(x − 1)2 (x2 + 2x + 2)
x−1
+
Cx + D
B
.
+
(x − 1)2 x2 + 2x + 2
x2 − 5x + 9 = A(x − 1)(x2 + 2x + 2) + B(x2 + 2x + 2)+
+ (Cx + D)(x − 1)2 ,
x2 − 5x + 9 = (A + C)x3 +
+ (A + B − 2C + D)x2 + (2B + C − 2D)x + (−2A + 2B + D).
x
⎧
x3
⎪
⎪
⎨ 2
x
x
⎪
⎪
⎩ :
⇒
⇒
⇒
⇒
A + C = 0,
A + B − 2C + D = 1,
2B + C − 2D = −5,
−2A + 2B + D = 9.
7
21
7
A = − , B = 1, C = , D = .
5
5
5
!
$
x2 − 5x + 9
dx =
(x − 1)2 (x2 + 2x + 2)
$
$
$
7
dx
dx
x+3
7
=−
+
dx.
+
5
x−1
(x − 1)2 5
x2 + 2x + 2
I=
"
#!$
1 2
(x + 2x + 2) = t ⇒ t = x + 1, x = t − 1; dx = dt.
2
% "
$
$
$
$
x+3
t+2
t dt
dt
dx
=
dt
=
+
2
=
x2 + 2x + 2
t2 + 1
t2 + 1
t2 + 1
1
1
= ln(t2 + 1) + 2 arctg t + C = ln(x2 + 2x + 2)+
2
2
+ 2 arctg(x + 1) + C.
% &
1
7
7
14
+ (x2 + 2x + 2) +
arctg(x + 1) + C.
I = − ln |x − 1| −
5
x − 1 10
5
''( I =
$
2x + 2
dx.
(x − 1)(x2 + 1)2
Bx + C
Dx + E
A
2x + 2
+ 2
+ 2
=
.
(x − 1)(x2 + 1)2
x−1
x +1
(x + 1)2
2
2x + 2 = A(x + 1)2 + (Bx + C)(x − 1)(x2 + 1) + (Dx + E)(x − 1)
2x + 2 = (A + B)x4 + (C − B)x3 + (2A + D + B + C)x2 +
+ (E − D + C − B)x + (A − C − E).
!
⎧
A + B = 0,
⎪
⎪
⎪
⎪
⎨ C − B = 0,
2A + D + B + C = 0,
⎪
⎪
E−D+C −B =2
⎪
⎪
⎩ A − C − E = 2.
"# A = 1 B = −1 C = −1 D = −2 E = 0$
%
$
2x
1
x+1
− 2
dx =
I=
−
x − 1 (x + 1)2 x2 + 1
$
$
$
$
d(x2 + 1) 1
d(x2 + 1)
dx
dx
−
−
=
−
=
2
2
2
2
x−1
(x + 1)
2
x +1
x +1
1
1
− ln(x2 + 1) − arctg x + C.
= ln |x − 1| + 2
x +1 2
& $
I=
''$(
I =
''$)
$ ''$*+
I =
2x + 5
dx.
x2 + 2x + 5
$
$
dx
.
x+3
dx
.
(x − 2)5
x+1
dx.
I=
x2 + 4x + 5
$
dx
.
I =
x3 − 2x2 + x
$
x dx
I =
.
(x − 1)(x + 1)2
$
dx
I =
.
(x3 − 1)2
$
x dx
I =
.
(5x2 + 2x + 4)2
$
!
" #
$%
& ! ' #
(
$
) *
R(sin x, cos x) dx,
R(sin x, cos x) + $%
! x2 y 3 + $% x y,
2
sin
x cos3 y + $% √ sin√
x cos y, ,
√ 2 √
( 2) ( 5 5)3 + $% 2 5 5.
-' ' ! $%
% x . '
#
% (
2 tg x2
1 − tg2 x2
sin x =
,
cos
x
=
.
1 + tg2 x2
1 + tg2 x2
/0 % ". $ % (
2t
x
1 − t2
tg = t, sin x =
, cos x =
,
2
2
1+t
1 + t2
dx =
2dt
.
1 + t2
) *
#
sinn x cosm xdx
2k + 1 (k 0, k ∈ Z)
m = 2k + 1
cos x = t
!"
$
m,
n
sin x = t
!
n = 2k + 1
#
##
2k + 1
)
I=
%& '
#
sin2 x cos3 x dx.
$
*
$
$
I = | sin x = t, cos xdx = dt| =
(1 − t2 )t2 dt =
$
(t2 − t4 )dt =
sin3 x sin5 x
t3 t5
− +C =
−
+ C.
3
5
3
5
=
* #
$
## $
$
I = sin2 x(1 − sin2 x) cos xdx = sin2 x(1 − sin2 x)d sin x =
$
sin3 x sin5 x
−
+ C.
= (sin2 x − sin4 x)d sin x =
3
5
(
)
%& %
$
+
cos x
$
I=
cos5 x
√
dx.
3
sin x
## $
cos xdx = d(sin x),
$
cos 4 x = (1 − sin2 x)2 .
##
(
"
$
1 − 2 sin 2 x + sin 4 x
d(sin x) =
I=
sin 1/3 x
$
$
−1/3
= (sin x)
d(sin x) − 2 (sin x)5/3 d(sin x)+
$
3
3
+ (sin x)11/3 d(sin x) = (sin x)2/3 − (sin x)8/3 +
2
4
3
(sin x)14/3 + C.
+
14
! "# $ % & &' (
) &
sin2 x =
sin 2x
1 − cos 2x
1 + cos 2x
, cos2 x =
, sin x cos x =
.
2
2
2
+
!!
*#
! &' ,
I=
#
sin4 x cos2 x dx
- . $
$
1 − cos 2x sin2 2x
·
dx =
sin2 x(sin x cos x)2 dx =
2
4
$
$
1
1
=
sin2 2xdx −
sin2 2x cos 2xdx.
8
8
I=
+ !/ &&
+% &
0 ! / && ,
$
$
1 − cos 4x
1
1
dx −
sin2 2xd(sin 2x) =
8
2
16
1
1
1
sin 4x −
sin3 2x + C.
= x−
16
64
48
I=
1 !/
$ % & cosk x0
(sink x)0 k = |m+n|
∈ N 0 !! )) 1 (
2
# ! 0 m n 1
1 0 !
$
$
I=
dx
.
cos3 x sin x
$
dx
tg2 x + 1
cos2 x
=
d tg x =
I=
sin x cos x
tg x
$
$
tg2 x
d tg x
=
+ ln | tg x| + C.
= tg x d tg x +
tg x
2
I=
$
dx
.
sin3 x cos5 x
dx
= d tg x,
cos2 x
3
1 + tg2 x
1
1
=
.
=
3
tg x
sin3 x cos3 x
tg x
1
√
·√
2
2
1+tg x
1+tg x
3
$
$
1
1 + 3 tg2 x + 3 tg 4 x + tg 6 x
1 + tg2 x
d tg x =
I=
d tg x =
tg x
3
tg3 x
$
3
1
tg −3 x +
+ 3 tg x + tg 3 x d tg x =
=
3
3 tg x
3
1
1
= ctg 2 x + 3 ln | tg x| + tg 2 x + tg 4 x + C.
2
2
4
! "# $%&'" %( $ &
) * !) $!
+ R(sin x, cos x) & %! $ % sin x − sin x $%
$! cos x = t.
+ R(sin x, cos x) & %! $ % cos x − cos x $%
$! sin x = t.
+ R(sin x, cos x) && $ ) % sin x
− sin x, cos x − cos x, $ && $! tg x = t.
$
dx
, I =
.
a2 cos2 x + b2 sin2 x
!
"
tg x = t =⇒ x = arctg t =⇒ dx =
#
t
1
sin x = √
, cos x = √
,
2
1+t
1 + t2
t2
1
, cos2 x =
.
sin2 x =
1 + t2
1 + t2
tg x = t,
$
$
I=
=
dt
.
1 + t2
dt
1+t2
a2
1+t2
+
b2 t2
1+t2
$
=
dt
1
=
a2 + b2 t2
b
bt
1
1
arctg + C =
arctg
ab
a
ab
%&!'
$
bdt
=
a2 + (bt)2
b
tg x + C.
a
$
I=
dx
.
sin 3 x cos2 x
$ sin x − sin x
cos x = t =⇒ sin x =
√
dt
1 − t2 =⇒ x = arccos t =⇒ dx = − √
.
1 − t2
(
$
I=−
dt
=−
√
2
1 − t (1 − t2 )3 t2
) *
$
dt
.
(1 − t2 )2 t2
+! +
1
(1 − t2 )2 t2
, +
I=
cos x
1
3 x
−
+ ln tg + C.
cos x 2 sin2 x 2
2
#
tg n xdx
1
−1
cos2 x
n
tg x = t
x = arctg t
# 4
I = tg 2xdx
tg 2 x =
!
"
dx =
#$
% & ' ( tg 2x = t" ) x = arctg t dx =
*+
1
2
dt
1+t2
1 dt
2 1+t2
$ 4
$
t dt
1
1
1
=
t2 − 1 + 2
dt =
I=
2
t2 + 1
2
t +1
t 1
tg3 2x tg 2x 1
t3
−
+ arctg tg 2x + C =
= − + arctg t + C =
6
2 2
6
2
2
tg3 2x tg 2x
−
+ x + C.
=
6
2
#
ctg n xdx
#
R(tg x) dx
tg x = t
t
$
tg x + 3
dx.
I = tg
x−1
, )( - )
. + ( !
/ )
##
% & . ! !
t = tg x =⇒ x = arctg t =⇒ dx =
( )
dt
,
1 + t2
( ) $ / /
I=
t+3
dt.
(t − 1)(t2 + 1)
% ) /&- /
t+3
A
Bt + C
=
+ 2
.
(t − 1)(t2 + 1)
t−1
t +1
t + 3 = A(t2 + 1) + (Bt + C)(t − 1).
t = 1 A = 2
! t2 ! ! B = −2
C = −1.
" #
2t + 1
2
−
dt =
I=
t − 1 t2 + 1
$
$
$
dt
d(t2 + 1)
dt
=2
−
−
=
t−1
t2 + 1
t2 + 1
= 2 ln |t − 1| − ln(t2 + 1) − arctg t + C.
$
" !
1
tg 2 t + 1 =
arctg tg t = t,
cos2 t
I = 2 ln | tg x − 1| + ln | cos x| − x + C.
$ ! % &
& % $ ! & ! & !
%
'( ) I = # cos 3x cos 9x dx.
* + #
$
$
1
1
(cos(−6x) + cos 12x) dx =
(cos 6x + cos 12x) dx =
I=
2
2
1
1
sin 6x +
sin 12x + C.
=
12
24
'( ,
I=
#
sin 2x cos 5x sin 9x dx.
$
$
1
1
(− sin 3x + sin 7x) sin 9x dx = −
sin 3x sin 9x dx+
I=
2
2
$
$
$
1
1
1
+
sin 7x sin 9x dx = −
(cos 6x − cos 12x) dx +
(cos 2x−
2
4
4
1 sin 12x sin 6x sin 2x sin 16x
− cos 16x) dx =
−
+
−
+ C.
4
12
6
2
16
#
sinn x cosm x dx
!
" #
$%& I = # sin2 x cos7 x dx.
$
$
I = sin2 x cos6 x cos x dx = sin2 x(1 − sin2 x)3 d sin x =
$
= (sin2 x − 3 sin4 x + 3 sin6 x − sin8 x) dx =
=
sin3 x 3 sin5 x 3 sin7 x sin9 x
−
+
−
+ C.
3
5
7
9
$%$
$
I=
sin5 x
√
dx.
cos x
' "
(
− sin x dx = dt
cos x = t!
$
$
−1/2
(1 − t2 )2
√
+ 2t3/2 − t7/2 dx =
dt =
−t
t
√
√
4 2√
2 √
4
2
= −2 t + t t − t4 t + C = cos x(−2 + cos2 x − cos4 x) + C.
5
9
5
9
I=−
) * +!
! "#
$%% I = # cos4 x dx.
2
1 + cos 2x
,
cos x = (cos x) =
2
$
$
1
1
(1 + cos 2x)2 dx =
(1 + 2 cos 2x + cos2 2x) dx.
I=
4
4
4
2
2
cos2 2x =
1+cos 4x
2
$
1 + cos 4x
1
1 + 2 cos 2x +
dx =
4
2
$
1
3
1
1 3
1
=
( + 2 cos 2x + cos 4x) dx = ( x + sin 2x + sin 8x) + C.
4
2
2
4 2
8
I=
I=
$
#
cos2 3x sin4 3x dx.
$
sin2 6x 1 − cos 6x
dx =
4
2
$
$
1
1 − cos 12x
1
=
(sin2 6x − sin2 6x cos 6x) dx =
−
8
8
2
1 x sin 12x sin3 6x
− sin2 6x cos 6x dx =
−
−
+ C.
8 2
24
18
I=
(cos 3x sin 3x)2 sin2 3x dx =
! " # $ % &
'm + n = 2k k ∈ N
( ) (# m n (# "# ( &
" # ( #
$
*
I=
3
cos2 x
dx.
sin8 x
+ ,
m = − 83
2
n = 3
m + n = − 83 + 23 = −2
$ ctg x = t )$
dx
− 2 = dt
sin x
3
2
cos2 x
= ctg 3 x.
2
sin x
$
I=−
2
3 5
3
t 3 dt = − t 3 + C = − ctg x 3 ctg 2 x + C.
5
5
n tg x ctg x
I = # tg4 x dx.
! " #
dt
=
I = tg x = t, x = arctg t, dx = 2
t + 1
$
$ 4
1
t dt
=
dt =
=
t2 − 1 + 2
t2 + 1
t +1
tg3 x
t3
− tg x+
= − t + arctg t + C =
3
3
3
tg x
+ arctg(tg x) + C =
− tg x + x + C.
3
$
I=
#
ctg 5 x dx.
! " # %& '
1
dx
ctg 2 x =
−1
= − d ctg x :
2
2
sin
x
sin
$x
$
1
−
1
dx =
I = ctg 3 x ctg 2 x dx = ctg 3 x
sin2 x
$
$
ctg 4 x
−
= − ctg 3 x d ctg x − ctg 3 x dx = −
4
$
$
1
ctg 4 x
− ctg x
+ ctg x d ctg x+
− 1 dx = −
2
4
sin x
$
ctg 4 x ctg 2 x
+
+ ln | sin x| + C.
+ ctg x dx = −
4
2
() ) " %
*
$
+, I = sindx3 x .
x
2t
dt
=
I = tg = t, sin x =
, dx =
2
1 + t2
1 + t2
$
$
1
(1 + t2 )2
1
1
2
=
dt =
+ + t dt =
4
t3
4
t3
t
1
t2
1
− 2 + 2 ln |t| +
+C =
=
4
2t
2
x 1
x
1
x
1
− ctg2 + ln | tg | + tg2 + C.
8
2 2
2
8
2
#
cosdx x
5 π
5
cos x sin 2 + x
5
! ""
$
I=
5 + 6 sin x
dx.
sin x(4 + 3 cos x)
#$ tg x2 = t
$
I=
2t
1+t2
5 + 12t2
2 dt
1+t 2
=
3(1−t ) 1 + t2
4 + 1+t2
$
5t2 + 12t + 5
dt.
t(7 + t2 )
%
5t2 + 12t + 5
A Bt + C
= +
.
t(7 + t2 )
t
7 + t2
#
5t2 + 12t + 5 = A(7 + t2 ) + t(Bt + C);
5
30
A = , B = , C = 12.
7
7
&
30
t + 12
5
dt =
+ 7
7t
7 + t2
5
15
12
t
= ln |t| +
ln(7 + t2 ) + √ arctg √ + C =
7
7
7
7
12
x
x
1
x
5
ln | tg | + 3 ln(7 + tg2 ) + √ arctg( √ tg ) + C.
=
7
2
2
7
7 2
$
I=
I=
I=
I=
I=
I=
I=
#
#
#
#
#
#
$
sin 6x cos 2x dx.
cos 2x cos 3x dx.
sin5 x cos2 x dx.
√
3
cos5 x sin2 x dx.
sin4 x dx.
sin4 x cos6 x dx.
dx
.
sin4 x cos6 x
$
sin x
dx.
I=
cos9 x
$
I = ctg5 x dx.
I=
$
I=
I=
$
tg8 x dx.
dx
.
sin5 x
! " #
$
√
#
R(x; n ax + b) dx
√
√
% R(x; n ax + b)dx
$ R(x; n ax + b) &
√
x n ax + b n
#
' $ "
(
√
ntn−1
tn − b
, dx =
dt, ax + b = t.
ax + b = tn , x =
)*+,a
a
. $
n
$
$
√
n
R(x; ax + b)dx = R
n−1
nt
tn − b
;t
dt.
a
a
% "
$
*+,
' ( / (
x + 1 = t2 , x = t2 − 1
/0 (
$
I=
t · 2tdt
=2
t2 − t − 1
# t+1
$
√
x+1
√
dx
x− x+1
$
I=
t2 dt
=2
2
t −t−1
dx = 2tdt.
$
1+
t+1
t2 − t + 1
dt.
% t −t−1 " 0
" #$
1 2
1
(t − t − 1) = z, t = z +
dt = dz.
2
2
1 ' ' (
2
$
$
$
z + 32
d(z 2 − 54 )
dz
dz
=
2t
+
+
3
=
5
5
2
2
2
z −4
z −4
z − 54
√
z − 5
3
5
2
2
√ + C = t + ln |t − t − 1|+
= t + ln z 2 − + √ ln
4
5 z + 25
√
t − 1 − 5
3
2
2
√ +C =
+ √ ln
5 t − 12 + 25
√
2 x + 1 − 1 − √5
√
√
3
√ + C.
= x + 1 + ln |x − x + 1| + √ ln √
5 2 x + 1 − 1 + 5
I = 2t + 2
$
R x;
n
ax + b
cx + d
dx
ax + b
= tn .
cx + d
I =
$
x
x−1
dx
x+2
! "
x−1
= t2 ,
x+2
" #
$
I=
1 + 2t2
x−1
= t, x =
x+2
1 − t2
6t
1 + 2t2
t
dt = −6
1 − t2 (1 − t2 )2
$
dx = (1 −6tt2)2 dt.
2t4 + t2 dt
.
(t2 − 1)3
$ $ ! %
2t4 + t2
2t4 + t2
=
=
2
3
(t − 1) )
(t − 1)3 )(t + 1)3
B
E
C
D
F
A
+
+
+
+
+
.
=
t − 1 (t − 1)2 (t − 1)3 t + 1 (t + 1)2 (t + 1)3
2t4 + t2
=
A(t − 1)2 (t + 1)3 + B(t − 1)(t + 1)3 + C(t + 1)3 +
(t − 1)3 )(t + 1)3
+ D(t − 1)3 (t + 1)2 + E(t − 1)3 (t + 1) + F (t − 1)3 / (t − 1)3 (t + 1)3 .
t = 1 : 3 = 8C ⇒ C = 3/8
t = −1 : 3 = −8F ⇒ F = −3/8
"##$
!
2t4 + t2 = A(t2 − 1)(t + 1) + B(t2 − 1)(t + 1)2 + C(t + 1)3 +
+ D(t2 − 1)2 (t − 1) + E(t2 − 1)(t − 1)2 + F (t − 1)3 =
= A(t4 − 2t2 + 1)(t + 1) + B(t2 − 1)(t2 + 2t + 1) + C(t3 + 3t2 + 3t + 1)+
+ D(t4 − 2t2 + 1)(t − 1) + E(t2 − 1)(t2 − 2t + 1) + F (t3 − 3t2 + 3t − 1).
%
2t4 + t2 = A(t5 + t4 − 2t3 − 2t2 + t + 1) + B(t4 + 2t3 − 2t − 1)+
+ C(t3 + 3t2 + 3t + 1) + D(t5 − t4 − 2t3 + 2t2 − t − 1)+
+ E(t4 − 2t3 + 2t − 1) + F (t3 − 3t2 + 3t − 1).
"##$ & & !
&
⎧
t5 :
⎪
⎪
⎪
⎨ t4 :
⎪ t3 :
⎪
⎪
⎩
t:
A + D = 0,
A + B − D + E = 2,
−2A + 2B + C − 2D − 2E + F = 0,
A − B + C − D − E − F = 0.
" C = 38 F
= − 38
⎧
A + D = 0,
⎪
⎪
⎪
⎨ A + B − D + E = 2,
⎪ −A + B − D − E = 0,
⎪
⎪
⎩
A − B − D − E = − 34 .
' A =
5
;
16
B =
11
;
16
5
D = − 16
; E =
11
16
$
t8 dt
1
= 12
t7 + t6 + t5 + t4 + t3 + t2 + t + 1 +
dt =
t−1
t−1
t8 t7 t6 t5 t4 t3 t2
= 12
+ + + + + + + t + ln |t − 1| + C.
8
7
6
5
4
3
2
√
! t = 12 x + 1
√
3
12
12
(x + 1)2
(x + 1)7
(x + 1)5
x+1
+
+
+
+
I = 12(
8√
7
6
5
√
√
3
4
6
√
√
x+1
x+1
x+1
+
+
+ 12 x + 1 + ln | 12 x + 1 − 1|) + C.
+
4
3
2
√
#
R(x; Ax2 + Bx + C) dx
" 12 (Ax2 + Bx + C) = t #
$
12
$ %
√
√
√
a2 − t2 , t2 − a2 , a2 + t2 .
&'(')
* # # $ %
&'(') !
√
a2 − t2 =⇒ t = a sin z t = a cos z,
√
a
a
,
t2 − a2 =⇒ t =
t=
sin z
cos z
√
a2 + t2 =⇒ t = a tg z t = a ctg z.
'('
I =
+ , "
1 2
(x + 2x − 3) = t x + 1 = t,
2
,
$ √ 2
x + 2x − 3
dx
(x + 1)3
x = t − 1, dx = dt
$
$ √2
(t − 1)2 + 2(t − 1) − 3
t −4
I=
dt =
dt.
t3
t3
- $
t=
2
,
cos z
√
t2 − 4 = 2 tg z, dt =
2 sin z
dz.
cos2 z
$
$ √2
$
1
t −4
2 tg z 2 sin z
dz
=
sin2 z dz =
dt
=
t3
2
( cos2 z )3 cos2 z
$
1
sin 2z
1
1
(1 − cos 2z)dz =
z−
+ C = (z − sin z cos z) + C.
=
4
4
2
4
I=
2
2
2
, cos z = , z = arccos ,
cos z
t√
t
2−4
√
t
.
sin z = 1 − cos2 z =
t
t=
√
2 2 t2 − 4
1
+C =
arccos −
4
t
t2
√
1
2
2 x2 + 2x − 3
=
arccos
−
+ C.
4
x+1
(x + 1)2
I=
!
" " # $%
%
I =
&'
$
I=
$
dx
√
( 5 + 2x + x2 )3
1 2
(x + 2x + 5) = t dx = dt
= 2
2
3
x+2=t
x= t−2
(5 + 2x + x )
dx
=
$
t = 2 tg z
dt
dt = cosdz2 z 1
√
=
=
=
2
4 cos zdz =
t2 + 4 = cos z
(t2 + 4)3
z
1
t
1
1
+C =
= sin z + C = sin arctg + C = 2
4
4
2
4 z2 + 1
$
4
x+1
+ C.
= √
4 5 + 2x + x2
$
dx
√
x>m
(x − m) ax2 + bx + c
dt
1
x − m = , dx = − 2 .
t
t
$
I = √ 2 dx
x 5x − 2x + 1
! " # $ %#
$
d(t − 1)
dt
√
=−
=
2
1
5
2
t − 2t + 5
(t − 1)2 + 4
−
+
1
2
t
t
t
1
√
1
2
2
+
5
+
= − ln |t − 1 + t + 2t + 5| + C = − ln − 1 +
+ C.
x
x2 x
$
I=
− dt
t2
$
=−
& ' #
$
xm (a + bxn )p dx,
m, n p ( ' %
) % *+ % + ,
% , -
. p ( % / , % +% a + bxn 0
p xm %
( % / , %
1 m+1
n
a + bxn = ts , s ( p
+ p ( % / , %
m+1
n
−n
ax + b = ts , s ( p
2
$
3
√
1+ 4x
√
dx.
x
$
1/3
x−1/2 1 + x1/4
dx.
I=
1
1
m+1
1
=⇒
= 2 −
m=− , n= , p=
2
4
3
n
1+x
1/4
= t3 =⇒ t =
3
.
1 + x1/4 =⇒
−1/2
=
x1/4 = t3 − 1 =⇒ x = (t3 − 1)4 =⇒ x−1/2 = (t3 − 1)4
1
=⇒ dx = 4(t3 − 1)3 3t2 dt = 12(t3 − 1)3 t2 dt
= 3
(t − 1)2
$
$
t · 12(t3 − 1)3 t2 dt
=
12
(t3 − 1)2
$
t3
−
= 12 (t6 − t3 )dt = 12t4
7
I=
t3 (t3 − 1)dt =
1
+ C.
4
!
t=
I = 12(1 +
√
4
3
x)
3
1+
1+
√
4
x
√
4
x,
√
1+ 4x 1
+ C.
−
7
4
" #$ %& %& '
& %& %& ( !
' ) %& **
( + ,#$*-. ,#$*#.*
' u = u(x; y) !
( ' u = Ref (z)
f (z) = u(x; y) + iv(x; y)* /% ( '
v = v(x; y) = Imf (z) '( f (z)*
∂u
=
⇒v =
1 ∂v
∂y
∂x
ϕ(x) x
2 !" """
# ∂u
∂v
∂u
=
dx + ϕ (x) = − " #"
∂x
∂x
∂y
∂u
dx + C
∂y
x
$" % & ' ϕ(x)
# ∂u
dx + ϕ(x)
v = Imf (z) =
∂x
ϕ(x) = −
#
x
# ∂u
dx
∂x
# ∂u
dy+ϕ(x)
∂x
ϕ (x) "
+
v(x) ( %"
)% '" *&" " % *
"% ( ! ( %"
+, u(x; y) = x3 − 3xy2
f (z)
f (z) f (0) = 0
- . / *& %" u(x; y) =
" % ( 0" '" *&" ("
" ( %" 1 "% ( u(x; y) = x3 − 3xy2 =
= x3 −3xy 2
#
∂v
∂u
=
= 3x2 − 3y 2 ⇒ v(x; y) = 3 (x2 − y 2 )dy + ϕ(x) =
∂y
∂x
∂v
∂u
2
3
= 6xy + ϕ (x) = −
= 6xy ⇒ ϕ (x) = 0 ⇒
= 3x y − y + ϕ(x)
∂x
∂y
⇒ ϕ(x) = C
" v(x; y) = Imf (z) = 3x2y − y3 + C f (z) = x3 − 3xy2 + i(3x2y −
− y 3 + C) "" C # % f (0) = 0 "
z = x + iy = 0 ⇒ x = y = 0 f (0) = 0 + iC = 0 ⇒ C = 0
2 %" f (z) = x3 − 3xy2 + i(3x2y − y3 )
= Ref (z) ⇒
" 3 " ' *4 %
% "% " ! " %!"
"
. " " " % 1%" '
" " & '" * % 0"
& "
$
$
$
2
ex dx,
√
sin x
dx,
x
x cos x dx
! "
#$ %
$
I=
sin x cos x dx.
& ' ( )
sin x cos x *
$
1
sin x d(sin x) = sin2 x + C,
2
$
1
I = − cos x d(cos x) = − cos2 x + C,
2
$
$
1
1
1
sin 2x dx =
sin 2x d(2x) = − cos 2x + C.
I=
2
4
4
I=
#$ +
!
1
1 2
sin x, − cos2 x
− 41 cos 2x
2
2
" # #
, ' . "
'
) " .
$
−6x2 − 5x + 8
dx
x3 + 2x2 − 1
## # ' /0
1
$
3x2 + 4x
dx = ln |x3 + 2x2 − 1| + C
x3 + 2x2 − 1
! "# "$
% && ' (
! ) * + ) , - %
. &/0&$
. 1
1 1
"2 &
+%
$
3
I=
3
I = |x − 2 = t6 , dx = 6t5 dt| =
√
3
x−2
dx.
√
(x − 2)2 − x − 2
$
t4
t2
6t5 dt =
− t3
1
dt =
=6
t +t +t+1+
t−1
t4 t3 t2
=6
+ + + t + ln |t − 1| + C =
4
3
2
√
√
3
3
(x − 2)2
x−2
x−2
+
+
+
=6
4
3
2
$
t4
dt = 6
t−1
$
3
2
√
√
+ 6 x − 2 + ln | 6 x − 2 − 1| + C.
"2 &
√
$
$
3
(x − 2)2 − x − 2
√
I=
dx
=
(x − 2)1/3 −
3
x−2
3
6
−(x − 2)1/6 d(x − 2) = (x − 2)4/3 − (x − 2)7/6 + C.
4
7
ax+b
cx+d
$
I=
5 − 3x
dx.
4 + 7x
! " # $ 5−3x
= t2
4+7x
&
%
x dx#
5 − 4t2
⇒
7t2 + 3
2
2
−8t(7t + 3) − 14t(5 − 4t )
−94t
⇒ dx =
dt =
dt.
2
2
(7t + 3)
(7t2 + 3)2
5 − 3x = t2 (4 + 7x) ⇒ x =
$'
$
$
−94t
t2 dt
dt
=
−94
=
2
2
(7t + 3)
(7t2 + 3)2
2 $ 2 3 3
$
$
t +7−7
dt
1
dt
1
3
dt
=
−94
·
−
.
= −94 ·
7
49
7
(t2 + 37 )2
t2 + 37
(t2 + 37 )2
I=
t
$ ( ) * #
$
I=
dt
7
3 2 =
2
3
(t + 7 )
1
2
$
dt
t
3 +
2
2
t +7
2(t + 37 )
.
+ ' ( , ,
( (
√
(5 − 3x)(4 + 7x) 47 21
7 5 − 3x
−
arctg
+ C.
I=
7
147
3 4 + 7x
-
$
I=
3
dx
(x − 1)2 (x + 2)
.
3
(x − 1)2 (x + 2) =
(x−12 )
(x
(x+2)2
2
+ 2)3 = (x + 2) 3 x−1
.
x+2
x − 1
dx
=
= t3 ,
I=
x+2
3
x−1 2
(x + 2)
x+2
$
dt
2t3 + 1
9t2 dt
=
3
x=
,
dx
=
.
3
3
2
1−t
(1 − t )
1 − t3
$
1
A
Bt + C
1
=
+
,
=
1 − t3
(1 − t)(t2 + t + 1)
1 − t t2 + t + 1
2
1
A = B = , C = ,
3
3
√
1
2t + 1
I = − ln |1 − t| + ln(t2 + t + 1) + 3 arctg √ + C.
2
3
1
I = − ln
2
√
√
√
√
3
3
√
x+2− 3x−1
23x−1+ 3x+2
√ √
+ C.
+ 3 arctg
3
33x+2
! " # !
$" %
! & !'
()'(
$
I=
√
!
dx
.
2x2 + 3x + 7
1
t − 32
1
3
2
, dx = dt =
I = (2x + 3x + 7) = t, 2x + = t, x =
2
2
2
2
$
1
dt
1
47
=√
= √ ln t + t2 + + C =
4
2
2
t2 + 47
4
√
1
= √ ln 4x + 3 + 2 4x2 + 6x + 14 + C.
2
$
√
I=
3x − 7
dx.
5x2 + 8x + 1
1
dt
t−4
2
, dx =
=
I = (5x + 8x + 1) = t, 5x + 4 = t, x =
2
5
5
$
3
(t − 4) − 7
5
dt =
2
8
5 5 t−4
+
(t
−
4)
+
1
5
5
$
$
2
d(t − 11)
47
dt
3
√
√
− √
=
= √
2
2
10 5
t − 11
5 5
t − 11
√
3 √
47
= √ t2 − 11 − √ ln |t + t2 − 11| + C =
5 5
5 5
√
3
47
=
5x2 + 8x + 1 − √ ln |5x + 4 + 5(5x2 + 8x + 1)| + C.
5
5 5
$
dx
√
(x + d) ax2 + bx + c
x + d = 1t
$
dx
√
.
I =
x x2 + 3
=
$
1
− dt/t2
dt
√
I = x = , dx = − 2 =
=
t
t
3t2 + 1/t2
√
$
$
dt
1
d( 3t)
√
=− √
= −√
=
3t2 + 1
3
( 3t)2 + 1
√
3 √x2 + 3
√
√
1
1
2
+
= − √ ln | 3t + 3t + 1| + C = − √ ln
+ C.
x
3
3 x
$
I=
dx
√
.
x2 7 − x2
$
$
dt
− dt/t2
1
t dt
√
I = x = , dx = − 2 =
=− √
=
t
t
7t2 − 1/t3
7t2 − 1
$
1 √
1
d(7t2 − 1)
1 7
√
= − 2 7t2 − 1 + C = −
=−
− 1 + C.
14
14
7 x2
7t2 − 1
$
dx =
#$
$
I=
I=
dx
√
.
(x2 + 9) x2 + 9
! x = 3 tg t"
3dt
cos2 t
x2 + 9 = 9(tg2 t + 1) =
1
3 dt/ cos2 t
=
3
27/ cos t
9
$
cos t dt =
9
.
cos2 t
1
sin t + C.
9
% sin t x
x
⇒ sin t = tg t · cos t =
3
tg t
x/3
x
=
=
=√
.
2
2
9 + x2
1 + (x/3)
1 + tg t
x
I = √
+ C.
9 9 + x2
x = 3 tg t ⇒ tg t =
#$
&
'(
''
√
6
x dx
√
√ .
3
x( x + 4 x)
$
3 − 4x
dx.
I=
9 − 5x
$
dx
.
I=
3
(x + 1)2 (x − 2)4
$
I=
$
I=
I=
I=
I=
I=
√
dx
.
11 + 5x + 6x2
√
2x + 5
dx.
7 + 8x − 11x2
$
$
$
$
dx
√
.
x 2x2 − 1
dx
√
.
x2 15 + 3x2
dx
√
.
(x2 − 5) x2 − 5
$
I=
x(3 + 4x3 ) dx.
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c c = xm
n
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f (x) dx −
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f (x) 0,
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λ → 0 f (x) dx
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f (x) dx
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a
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(f (x) − ϕ(x)) dx 0.
,
a
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f (x) dx −
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a
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f (x) dx
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[a; b]
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a
m M
f (x) [a; b] !" x ∈ [a; b]
m f (x) M.
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dx
m
a
&
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f (x) dx M
a
dx.
a
dx = b − a f (x) = 1
a
n
f (ξi )Δxi =
n
i=1
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i=1
#'
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m
b
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a
b−a
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) μ * * m M +
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*
*
m M $
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f (x) dx
a
b−a
= f (ξ)
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[a; b]
!
"
! f (x) # [a; b] $
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f (x) dx.
a
% a & '
& ' (&
' & x
b
! & )
'
& t * &
' x+
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I(x) =
f (t) dt.
a
1 2
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,-. /0
⎞
⎛ x
$
⎝ f (t) dt⎠ = f (x).
a
!
! " # #$ #%
x & Δx ' (
x+Δx
$
I(x + Δx) =
f (t) dt.
a
)
& I(x) (
x+Δx
$
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ΔI =
a
*
f (t) dt.
a
+ [a; x + Δx] " x
[a; x]
[x; x + Δx]
,-
.' ' (
x+Δx
$
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f (t) dt =
a
x+Δx
$
f (t) dt +
a
/
f (t) dt.
x
0' 1 / 2 *
x+Δx
$
ΔI =
f (t) dt.
#3
x
, ' " #3
" '
ΔI = f (ξ)Δx, ' ξ ∈ [x; x + Δx].
+ 4 " ' Δx(
ΔI
= f (ξ), ξ ∈ [x; x + Δx].
Δx
##
Δx → 0
!
" #$
% & '
$x
I(x) =
f (t) dt
a
( ") '
f (x)
* ( ' "
" " + F (x) , "
( f (x) I(x) = F (x) + C
$x
f (t) dt = F (x) + C.
-
a
+ x = a
$a
f (t) dt = 0,
a
C = −F (a).
+ C - . x = b
F (a) + C = 0
$b
f (t) dt = F (b) − F (a).
/
a
0 1) ,2( ' * $
f (x) [a; b]
% + $ "
& ( ")
$π/2
I=
sin x dx.
0
I=
π/2
− cos x|0
= − cos
π
+ cos 0 = 1.
2
$b
f (x) dx,
a
f (x) !" [a; b] #"$
x " t
x = ϕ(t).
% &
ϕ(α) = a,
ϕ(β) = b.
"
'"$
ϕ(t) ! ϕ (t) !"
[α; β](
)
! t α β ! #"$ ϕ(t) *
! !" a x b
+ * ,- #
!
$b
$β
f (x) dx =
a
f (ϕ(t))ϕ(t) dt.
% .&
α
/ F (x) 0 ! #"$
F (x) = f (x) 1 # 2, 34 0 $
f (x)
$b
f (x) dx = F (b) − F (a).
a
% 5&
F (x) x = ϕ(t) F (ϕ(t))
f (ϕ(t))ϕ(t).
(ϕ (t))t = (F (x))t = Fx · xt = f (x) · ϕ (t) = f (ϕ(t))ϕ(t).
! "#
$β
$
f (ϕ(t))ϕ(t) dt = F (ϕ(β)) − F (ϕ(α)) = F (b) − F (a).
%&'()'*
α
+ , %&'()-* %&'()'*
%&'().*(
+ (
&'(/
I=
$1
0
dx
√
2
x + 4x + 20
$1
dx
=
+ 4x + 20
0
1 2
(x + 4x + 20) = t,
x + 2 = t,
= 2
=
α = a + 2 = 2,
β =b+2=3
$3
√
√
dt
8
√
√ .
= ln t + t2 + 16 |32 = ln 8 − ln(2 + 20) = ln
=
2
t + 16
2 + 20
+ 0
I=
√
x2
2
&'(1
I=
#a √
0
a2 − x2 dx
x = a sin t,
x=0 ⇒ t=0
+ 0 I = dx = a cos t dt, x = a ⇒ t = π/2 =
$π/2
$π/2
$π/2
a2
a2 − a2 sin2 ta cos t dt = a2
cos2 t dt =
(1 + cos 2t) dt =
=
2
0
0
0
sin 2t π/2 πa2
a2
t+
.
=
=
2
2
4
0
u(x) v(x)
[a; b]
d(uv) = v du + u dv.
!" a b !
$b
$b
duv =
uv|ba
=
a
$b
v du +
a
u dv,
a
$b
$b
u dv =
a
uv|ba
−
v du.
#$%&'(
a
)! #$%&'( !*
!+ !
π
$%$ I = # x sin x dx
0
,
+
#π
0
u = x,
du = dx
I =
dv = sin x dx, v = − cos x
cos x dx = π + sin x|π0 = π.
= −x cos x|π0 +
y
A
y=f(x)
B
f(ξi )
ξ
O
x0 =a
i
x n =b
xi-1 x i
x
f (x) 0 [a; b]
Ox y = f (x) x = a
x = b !"" "
#! $% & !' ! ' (
) ! (
a = x0 < x1 < x2 < . . . < xn = b
( [a; b] *+! ,
[x0 ; x1 ], [x1 ; x2 ], . . . , [xn−1 ; xn ],
λ=
max
i=1,2,··· ,(n−1)
|xi+1 − xi |.
- . % +! ( " !/
" ξi
0 1 (!2" 2 % %
% +! f (x) &
f (ξi) ! (" /
!' +!" !""
[xi ; xi−1 ] !' ! % "
f (ξi ) * 34,
5
n
S ≈ f (ξ1 )Δx1 + f (ξ2 )Δx2 + . . . + f (ξn )Δxn =
i=1
f (ξi )Δxi.
λ → 0 (n → +∞)
S = lim
λ→0
n
$b
f (ξi )Δxi,
i=1
S=
f (x) dx.
a
! " #$
#b
f (x) dx
a
y = f (x) 0 Ox
x = a x = b
% " & % ! '
# # " % % % (") V = V (t) > 0 *$
' " !" t = α t = β
" " ! % % % ' ! $
(
S = V · t.
+,
% # - .
/ . " [α; β] Ot . $
" " t0 = α < t1 < t2 < . . . < tn = β
Δti = ti − ti−1
+ * "! " .
" τi
0 1 " ' . . & &
! "! " ! "$
-' - (") V (t)
. "
2
S ≈ V (τ1 )Δt1 + V (τ2 )Δt2 + . . . + V (τn )Δtn =
n
V (τi )Δti .
+
i=1
$
+
' 3 " $
"!
3 % % "
S = lim
n→+∞
n
i=1
$β
V (τi )Δti ,
S=
V (t) dt.
α
++
V (t)
$β
|V (t)| dt.
S=
α
! " # $
% $&'# $#
!
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$ n " $ "
ξi #
( % )* $ h
h=
+
,
b−a
.
n
x0 = a, x1 = a + h, x2 = f + 2h . . . , xi = a + ih.
- , $. f (x)
# . # #
f (a + h), f (a + 2h), . . . , f (a + nh).
/ , 0 # $ h
$$ # 1 $ "$& $$
Sn = f (a + h) · h + f (a + 2h) · h + . . . + f (a + nh) · h =
n
f (a + ih).
=h
i=1
%
#b
a
x dx (a < b, )
[a; b]
!
" # # q (q > 1)
x0 = a, x1 = aq, x2 = aq 2 , . . . , xn = aq n = b.
$ % qn = ab !
& '
Δx1 = x1 − x0 = a(q − 1), Δx2 = x2 − x1 = aq(q − 1), . . .
Δxi = xi − xi−1 = aq i−1 (q − 1).
$ ( f (x) = x )
a, aq, aq 2 , . . . , aq n−1 .
* ) + %
' ,
Sn = a2 (q − 1) + (aq)2 (q − 1) + . . . + (aq n−1 )2 (q − 1) =
= a2 (q − 1)
n−1
q 2i .
i=0
-
,
S=
( ab )2 − 1
q 2n − 1
=
q2 − 1
q2 − 1
q → 1 n → +∞.
+
$b
a
2
b
−1
a2
= a2
x dx = a2 lim
(q
−
1)
2−1
q→1
q
n→+∞
q−1
b2
b2 − a2
=
.
−
1
lim 2
2
q→1 q − 1
a
2
. #' ') '
# )!
/ # % 0
% , ( f (x)
[a; b]
$b
x dx =
a
b
b2 − a2
x2
.
=
2 a
2
!
$1
I =
"# $
(x − 2)3 dx.
0
%
(x − 2)
1
I=
= 4 − 4 = −3,75.
4
0
4 1
√
I =
$3
"# "
1
%
"# &
dx
.
x2 + 1
√
√3
π
π π
I = arctg x1 = arctg 3 − arctg 1 = − = .
3
4
12
$1
.
I = x2dx
−4
0
%
1
=
I = 14 ln x−2
x+2 0
' ( )
#b
f (x) dx
1
ln 3 − ln 1 = 14 ln 13 = − 14 ln 3.
x = ϕ(t)
1
4
*+
)
)
a
x'
) (
) (
x=a
"# ,
t = α
t = β
x=b
$9 √
x dx
√ .
I=
x+2 x
1
%
)
x = t2 '
t
1 = t2 ⇒ t = 1, 9 = t2 ⇒ t = 3.
dx = 2t dt
$3
I=
1
$3
1−
=2
1
2t2 dt
=2
t2 + 2t
2
t+2
$3
1
t dt
=2
t+2
$3
1
t+2−2
dt =
t+2
dt = 2 (t − 2 ln |t + 2|) |31 = 4 1 − ln
I=
$10
3
dx
√
.
(x − 4) x + 6
x + 6 = t2
t
x = 3 t = 3,
x = 10 t = 4.
$4
I=
3
5
.
3
dx = 2t dt
t − √10 4
2t dt
1
√ =
= 2 √ ln
(t2 − 10) t
2 10 t + 10
3
4 − √10
3 − √10
1
√ − ln
√ =
=√
ln
10
4 + 10
3 + 10
√
4 − √10 3 + √10
1
7 + 2 10
1
√
√ = √ ln
= √ ln
.
3
10 4 + 10 3 − 10
10
I=
#2a √
2ax − x2 dx.
0
12 (2ax − x2) = t
a−x = t x = a−t dx = − dt 2ax−x2 = a2 −t2 x = 0 ⇒ t = a
x = 2a ⇒ t = −a
$a √
I =
a2 − t2 dt.
−a
t = a sin z dt = a cos z dz
= a cos z t = −a ⇒ z = − π2 t = a ⇒ z =
2
$π/2
I=a
π
2
√
a2 − t2 =
$π/2
cos z dz = a
(1 + cos 2z) dz =
2
2
0
−π/2
2
=a
π/2
πa2
sin 2z
.
=
z+
2
2
0
z
[− π2 ; π2 ]
!
" " "
#
$π/2
I=
$π/2
cos z dz = 2 cos2 z dz.
2
0
−π/2
!
$ "
# $
u = x;
du = dx
I =
dv = ex dx; v = ex
I=
#1
xex dx.
0
"
$1
= xex |1 − ex dx = e − ex |1 = 1.
0
0
0
"
#
%
$
$ "
I=
#1
x arctg x dx.
0
u = arctg x; du = 2dx
x +1
I =
2
dv = x dx; v = x2
=
=
1
$1 2
$1
x dx
π 1
1
x2
arctg x −
=
−
2
2
x2 + 1
8 2
0
0
1−
0
1
x2 + 1
dx =
π 1
π π 1
π 1
= − (x − arctg x)|10 = −
1−
= − .
8 2
8 2
4
4 2
f (x) [a; b]
n
ξi
y = ex [a; b] n
ξi
# x dx !
b
a
f (x) = x " [a; b] n
# ξi
I=
$1/2
√
0
! I =
#3
2
√
0
ex dx
$1
" I =
dx
1 − x2
dx
x2 + 4
$4 √
I=
1
x dx
√ .
1+ x
y
b
a x
0
1
b
S=
4
a
$a √
a2 − x2 dx.
0
1
S = |x = a sin t, x = 0 =⇒ t = 0, x = a =⇒ t = π/2| =
4
$π/2
$π/2
1 + cos 2t
b 2
1
1
πab
π/2
2
= ·a
dt = ab( t + sin 2t)|0 =
.
cos t dt = ab
a
2
2
4
4
0
0
! ! " #
S = πab.
! a = b = R
S = πR2 .
$ %! !
! % ! & $" ! !'
! ( ! ) " & * * +! ,
- & " $
a x b,
y (x) y (x).
. ! ! ( !
$b
(y (x) − y (x))dx.
SABCD =
a
+/0,
y
y=yb (x)
D
C
B
A
y=yH (x)
0
a
x
b
y (x) y (x) x ∈ [a; b].
! " #
x = x(t),
y = y(t),
$t2
S=
y(t)x (t) dt.
$%&'
t1
" ! $(&)*
y = f (x) = y(t), dx = x(t) dt& +" t1
, a, t2 - , b&
$%&'
x = a(t − sin t),
y = a(1 − cos t)
. # /, ,
!" t1 = 0 t2 = 2π & $0&
y
a
a
t=2π
t=0
x
$2π
$2π
a(1 − cos t)(a(t − sin t)) dt = a
(1 − cos t)2 dt =
2
0
0
$2π
$2π
1 + cos 2t
= a2 (1 − 2 cos t + cos2 t) dt = a2
1 − 2 cos t +
dt =
2
0
0
2π
3
sin 2t
2
t − 2 sin t +
= a2
= 3πa .
2
4
0
r = r(ϕ) ! "
OAB # $$%
A
r
+d
r
α
O
r=r(ϕ)
dϕ
r
B
β
r
dϕ
! " dS = 12 r2 dϕ,
1
S=
2
$β
#$%!&'
r2 dϕ.
α
$%!& r = cos 3ϕ
π/6
r
( )
S =2·
=
1
2
1
2
$π/6
cos2 3ϕ dϕ =
0
$π/6
(1 + cos 6ϕ) dϕ =
0
1
2
ϕ+
π/6
sin 6ϕ
π
= 12 .
6
0
" *+ V ! "
,
Ox.
x : S = S(x).
a b
[a, b] n
a = x0 < x1 < x2 < · · · < xn−1 < xn = b.
!
Ox.
"
n #$
% &
$
xi−1 xi Δvi . ' %
V = Δv1 + Δv2 + · · · + Δvn =
n
Δvi .
i=1
&
$ xi−1 xi . (% #$ &$ #
$
% % [x1 − xi−1
ξi ∈ [xi−1 , xi]
#$ %
)
* s(ξi )Δxi.
! #$ &$ *
V ≈
n
+,-,.
s(ξi )Δxi .
i=1
/ #$
%
+,-,. *
V = lim
λ→0
n
+,-0.
s(ξi )Δxi.
i=1
*
$b
V =
+,-1.
s(x) dx.
a
,-,
2
2
2
xa2 + yb2 + zc2
= 1.
2 *
)
Ox ($ x = d (d ∈ [−a; a]) !
x = d
√
z2
y2
+
− d2 )
b2
(a2
a2
a1 = ab
c2
(a2 − d2 )
a2
√
c
b1 = a a2 − d2
a2 − d2
=1
πbc 2
(a − d2 ).
a2
d = x (−a x a)
S = πa1 b1 =
!"
S(x) =
# $%&
!
πbc 2
(a − x2 ).
a2
'"
$a
V =
$a
S(x) dx = 2π
−a
0
bc
= 2π 2
a
bc 2
(a − x2 ) dx =
a2
a
x3
4
a x−
= πabc.
3 0 3
2
(
a = b = c = R
R
a b c
V = 43 πR3
) * ! ' * * Ox
+ * , - $./ *0
/ $! , -1 S .* x
! * / ! *!
S(x) = πy 2 .
)** . * ',
$b
Vx = π
a
y 2 dx.
2
y
y=f(x)
O
a
dS
b
x
y = (x − 1)2, x = 0, y = 0
x y y x
1
$1
π
(x − 1)5
4
Vx = π (x − 1) dx = π
= 5
5
0
0
$1
Vy = π
√
( y + 1)2 dy = π
0
1
y 2 4 3/2
17π
+ y + y =
2
3
6
0
! L " " A " B
# $ %& ' " A1 , A2 , . . . , An−1
(
! A = A0 ) B = An ! " )
* " + A0 A1 A2 . . . An−1 An
, L !
"
! # !
$ ! ! # %
-* ! ' AB
+ A0 A1 . . . An
y
B
A i-1
A1
A n-1
Ai
A
x
n
Δli ,
i=1
Δli Ai−1Ai
Δli =
(xi − xi−1 )2 + (yi − yi−1 )2 = Δx2i + Δyi2 .
[xi−1; xi]
f (xi ) − f (xi−1 ) = f (ξi )(xi − xi−1 )
yi − yi−1 = f (ξi )(xi − xi−1 )
Δx2i + Δyi2 =
Δyi = f (ξi )Δxi
Δx2i + f 2 (xi )Δx2i =
1 + f 2 (ξi )Δxi .
ξi [xi−1; xi]
!
"# $ "" ξi
% "#& " ξi '
"# (
n
Δli =
i=1
n
1 + f 2 (ξi )Δxi .
[a; b]
l = lim
1+
n→+∞
i=1
f 2 (ξi )Δxi
$b
=
1 + f 2 (x)
1 + y 2 dx.
a
!"#
1 + y 2 dx.
$%
"& & &"'
'
*
( & ) & x=x(t),
y=y(t)
dl =
dl =
dx2 + dy 2 =
x 2 dt2 + y 2 dt2 = x 2 + y 2 dt
& ) & # +"
$t2
l=
x 2 (t) + y 2 (t) dt.
t1
( # & $%
"
x = r cos ϕ,
y = r sin ϕ,
dx = (r cos ϕ)ϕ dϕ = (r cos ϕ − r sin ϕ) dϕ,
dy = (r sin ϕ)ϕ dϕ = (r sin ϕ + r cos ϕ) dϕ.
& & " " & $%* )
dl = r2 + r 2 dϕ,
& "' ' &
l=
$β
α
r2 + r 2 dϕ.
$$
√ y = ln x
3.
√
$ 3√
1 + x2
dx.
L=
x
1
x = tg t =⇒ dx =
$π/3
L=
π/4
=2−
dt
=
sin t cos2 t
√
$π/3
dt
,
cos2 t
sin2 t + cos2 t
dt =
sin t cos2 t
t π/3
1
+ ln tg =
cos t
2
π/4
π/4
√
π
= 0,92.
2 − ln
3 tg
8
$2π
l=a
(t − sin t) 2 + (1 − cos t) 2 dt =
0
$2π
(1 − cos t)2 + sin2 t dt =
=a
0
$2π
$2π
$2π
t
2 t
2(1 − cos t) dt = a
4 sin dt = 2a sin dt =
=a
2
2
0
0
0
2π
t
= −4a cos = −4a(−1 − 1) = 8a.
2 0
L
!" u = f (x; y) # $%& "
L
y = y(x) #'
n
f (ξi ; y(ξi ))Δli =
i=1
n
f (ξi ; y(ξi )) 1 + y 2 (ξi )Δxi.
i=1
Δli → 0 n → +∞
u = f (x; y)
!
$
$
f (x; y(x))dl =
L
f (x; y(x))dl =
(maxΔli →0)
AB
""# dl =
L
f (ξi ; y(ξi ))Δli,
i=1
1 + y 2 (x) dx
x = x(t) y = y(t)
x = r cos ϕ y = r sin ϕ
!
"!
$ %%& !
$
$tB
f (x; y) dl =
AB
tA
$
$ϕB
f (x; y) dl =
#
' %! (
f (x(t), y(t)) x 2 (t) + y 2 (t) dt
f (r cos ϕ; r sin ϕ)
r2 (ϕ) + r 2 (ϕ) dϕ,
'
ϕA
AB
n
lim
n→+∞
tA tB ϕA ϕB
A B
t
!
ϕ
$ %
x = R cos t y = R sin t A(0; R) B(R; 0)
" # f (x; y) = x · y
) *
( + %!
$
$tB
f (x; y) dl =
AB
xy
tA
x 2 (t) + y 2 (t) dt.
0 = R cos t
tB = π2 R = R cos t
tA = 0 xt = −R sin t yt = R cos t
π
$
$2
f (x; y) dl =
R cos t · R sin t R2 sin2 t + R2 cos2 t dt =
0
AB
π
= R3
$2
sin t cos t dt = R3
0
π
sin2 t 2 R2
.
=
2 0
2
A B
tA < tB
AB
!
" γ = f (x; y) # " f (x; y)
P (x; y)
$ AB
%& P " '
() * %+ !
Δmi Ai−1Ai ' ,-) Δmi ≈ f (xi; yi)Δli
%+ AB m ≈
n
≈
f (xi; yi )Δli *
!
i=1
,./ !
$
m=
f (x; y) dl.
',.01)
AB
2 f (x; y) = 1 ,.01
,.3
AB
,.3
√
y = x2 x ∈ [0; 2]
γ = 3x
4 $ ,.01 y = 2x
√
√
0
0
$2 √
$2
3 √
2
m = 3x 1 + 4x dx =
1 + 4x2 d(1 + 4x2 ) =
8
√
3 2 1
3 2
1 + 4x2 2 = (93/2 − 1) = 6,5.
= ·
8 3
4
0
y
B
dl
y=f(x)
A
O
a
dx
b
x
Ox
! !" # $
% & '()*
dσ = 2πy dl,
dx
$b
σ = 2π
y dl = 2π
a
$b
y 1 + y 2 dx.
&'( +,)
a
'( +-
. * / ! $ ! %
x2 + y2 = R2 Ox. 0 1"
% '( ,
y 2 = R 2 − x2 .
$R
V =π
−R
2
$R
y dx = π
−R
−R
R
4
(R2 − x2 )dx = πR3 .
3
!
x
y = − , 1 + (y )2 =
y
" #
x2
1+ 2 =
y
x2 + y 2
R
= .
y
y
$ %
$R
$R
R
2
σ = 2π y 1 + y ( )dx = 2π y dx =
y
−R
−R
2
= 2πRx|R
−R = 4πR .
& !
' ( (
) * +
I
t I = I(t) Q
T
& ) , & [0; T ] t0 = 0,
t1 , t2 , . . . , tn = T n # +
[t0 ; t1 ], [t1 ; t2 ], . . . , [tn−1 ; tn ].
-. Δti = ti − ti−1
/ 0 ! .
% % τi
1 2 $ Δti
* 3 I(t) . % % τi $
I(τi)
4 ( # $
)
*
$ ! #
ΔQi ≈ I(τi )Δτi,
Q≈
n
i=1
I(τi )Δτi .
Q
n
Q = n→+∞
lim
I(τi )Δτi .
λ→0
i=1
! λ " #$
% ! & '()
$T
Q=
I(t) dt.
0
*(+ L =
!" #$ %
, $ -
.
x / $ .
L − x (L − x)50
0 #
1
dx #
dA = −(L − x)50 dx.
-2 # !
$L
(L − x)2 L
A = −(L − x)50 dx = 50
=
2
0
0
= −25L2 = −25 · 1002 = −250000.
*+ & " %
'
*(3 (" %"
) V = t2 *" + %" $
,
,
$
$2
S=
0
2
t3
t dt = = 1.
8 0
2
√
y=2 x
y = 2x
y
y=2x
C 2
A
y=2 x
m
O
B
1
x
! "
Ox − OmAB OAB
$b
$1
(y − y )dx =
S=
√
2 x dx −
0
a
$1
0
#
1
1 1
4 3/2
2x dx = x − x2 = .
3
3
0
0
! "
#
Oy − OAC
OmAC $ % &
!
' ( " ) "
( "
$2
$2
(x − x )dy =
S=
0
0
y
dy −
2
$2
0
2
2
y2
y3
1
y2
dy = − = .
4
4 0 12 0 3
x = a cos t,
y = b sin t.
! " # "
$t2
S=
y(t)x (t) dt.
t1
y
t=π/2 b
-a
O
a
t=0
x
-b
$ % " " #&% ## ' (
) " " " y = b sin t, dx = d(a cos t) = −a sin t dt t ∈ [ π2 ; 0] #
$0
S = −4ab
sin2 t dt = −2ab
π/2
= −2ab(t −
$0
(1 − cos 2t) dt =
π/2
0
1
sin 2t) = πab.
2
π/2
$ # * # ! +
' " # , " # ,
-
! " r = a sin 2ϕ #
α = 0 β = π2
S=
1
2
$β
ρ2 dϕ =
a2
2
α
r = a sin 2ϕ
$π/2
sin2 2ϕ dϕ =
0
π/2
$π/2
1
a2
πa2
a2
.
(1 − cos 4ϕ) dϕ = (ϕ − sin 4ϕ) =
=
4
4
4
8
0
0
r
y = 2√x y = 2x Ox Oy
! " # $%
V2 & ' V1 ( ) * OmAB + V2 (
' OAB Ox
, ' +
V1
$1
Vx = V1 − V2 = π
4x dx − π
0
$1
2
4x dx = π
0
1
2x2 0
) Oy
# /% ! x y y x
$b
Vy = π
a
x2 dy.
1
2
x3
− 4 = π.
3 0
3
- .
$2
Vy = π
0
y2
dy − π
4
$2
0
y4
dy = π
16
2
2
4π
y 5
y 3
=
.
−
12 0 80 0
15
R
x2 +y2 = R2 Ox !" # $
√
x
R 2 − x2 , y = − ,
y
2
x2 + y 2
R
x
= .
1 + y2 = 1 + 2 =
y
y
y
y=
% $$ # $
$b
S = 2π y 1 + y 2 dx,
a
"
$R
S = 2π
y
−R
R
2
dx = 2πRx|R
−R = 4πR .
y
√
& y = ln x (x ∈ [1; 3])
' () ) )$
* + ,-
√
$ 3√
l=
1
$π/3
=
x = tg t
1 + x2
x =√1 ⇒ t =
dx =
dx = cosdt2 t x = 3 ⇒ t =
x
π/4
$π/3
=
π/4
$π/3
dt
=
sin t cos2 t
sin2 t + cos2 t
dt =
sin t cos2 t
π/4
sin t dt
+
cos2 t
$π/3
π/4
dt
=
sin2 t
t π/3
1
+ ln tg =
cos t
2
π/4
π
4
π
3
=
√
√
π
1
1
= 2 − 2 − ln 3 − ln tg = 2 − 2 −
2
8
2
√
√
1 3( 2 − 1)
≈ 0,91.
= 2 − 2 − ln √
2
2+1
1 − cos π4
ln 3 + ln
1 + cos π4
=
R
x = R cos t,
y = R sin t,
l=
#t2 2
x + y 2 dt =
t1
#2π
#2π
=
(R cos t) 2 + (R sin t) 2 dt = R dt = 2πR.
0
0
r = 1 − sin ϕ.
r = − cos ϕ!
r2 + r 2 = (1 − sin ϕ)2 + cos2 ϕ = 2(1 − sin ϕ) =
π ϕ
π ϕ
π
+ ϕ = 4 cos2
+
= 2 cos
+
.
= 2 1 + cos
2
4
2
4
2
" #!
$π/2
l=2
−π/2
2 cos
π
4
+
π ϕ π/2
ϕ
dϕ = 8 sin
+
= 8.
2
4
2 −π/2
$ % &' ( % % ' % '
* +
)
,
O
F = −kx k
! x " #
" $ "
x = a x = b
dx
dA
dA = −kx dx.
$b
F (x) dx = −
a
kx dx = −k
a
$b
A=
a b
k
x2
= (a2 − b2 ).
2
2
! y = 0
y = 4 − x
y = (x + 2)2
x = R cos3 4t ,
y = R sin3 4t .
"
r = a(1 + cos ϕ)
#
x = 0 y = 0 z = 0 x + y + z = 1
! x = const
!
y2 = x Ox x = C
$ "
y = x (y ∈ [0; 1])
Oy
%
y = 2x (x ∈ [0; 1]) Ox
& y = ax2 (a > 0)
# $ x
x = R cos3 4t ,
y = R sin3 4t .
I =
$+∞
1
dx
xα
$+∞
$b
dx
I=
= lim
x−α dx.
xα b→+∞
1
1
α = 1, ! ln x b → +∞ "
# !
$ α = −1
$b
+∞
1
I = lim
x−α dx = lim
% &'
.
b→+∞
b→+∞ (1 − α)xα−1 1
1
$ α < 1 # !
1
# !
$ α > 1 α−1
(& ) * +
y
α=1
y=1/xα
α1
,
" * !
.
/ 0 / )0*) % *)0 !
x > 1 . . * '
&
I =
$+∞
1
dx
1 + x2
!
$+∞
I=
1
dx
= lim
1 + x2 b→+∞
$b
1
b
dx
= lim arctg x =
2
b→+∞
1+x
1
= lim (arctg b − arctg 1) =
b→+∞
4
$+∞
I=
4
I=
π π
− .
2
4
$+∞
dx
√ = lim
x b→+∞
$b
4
π
.
4
dx
√ .
x
√
√
dx
√ = 2 lim x|b4 = 2 lim ( b − 2) = +∞.
b→+∞
b→+∞
x
!
" # $ #% & $ '
b
$
$
b
f (x) dx = lim
−∞
(
*
f (x) dx.
a→−∞
a
$
#% $ ) '
$+∞
$c
$+∞
f (x) dx =
f (x) dx +
f (x) dx,
−∞
−∞
c
+ , ) #
( * # '
,
$ * #
$+∞
I=
−∞
$0
I=
−∞
arctg x dx
+
1 + x2
$+∞
0
arctg x dx
.
1 + x2
arctg x dx
.
1 + x2
$0
−∞
arctg x dx
= lim
a→−∞
1 + x2
$0
0
arctg x dx
1
lim arctg 2 x =
=
2
1+x
2 a→−∞
a
a
=−
1
π2
lim arctg 2 a = − .
2 a→−∞
8
$+∞
0
arctg x dx
= lim
b→+∞
1 + x2
=
$b
b
2
arctg x dx
1
lim arctg x =
=
2
1+x
2 b→+∞
0
0
2
1
π2
lim arctg b = .
2 b→+∞
8
I =−
π2 π2
+
= 0.
8
8
!" #
!" $
$+∞
−∞
t = arctg x =⇒ dt =
' '(
dx
, x = −∞ =⇒
1 + x2
⇒ t = −π/2, x = +∞ =⇒ t = π/2.
$+∞
−∞
dx
=
1 + x2
dx
.
1 + x2
% &
'
)
$π/2
−π/2
π/2
dt = t
= π.
−π/2
$+∞
+∞
f (x) dx = F (x)
= F (+∞) − F (a),
a
a
F (+∞) = lim F (x).
x→+∞
f (x) [a, b)
b
$b
f (x) dx
a
!
" # $ % b % b − ε (ε > 0)
f (x) [a; b − ε]
$b−ε
& ' % (
f (x) dx
) *% +( '
$b
a
( ( ε → 0,
$b−ε
f (x) dx = lim
f (x) dx
ε→0
a
- ./
a
0 ! ( %
1 2 (2 ( $
- / & ( $ - 3
/
4
I =
$1
0
dx
.
x−1
1
5 6 , (
x−1
x = 1 7 % ( ! ( +
$1
$1−ε
1−ε
dx
dx
= lim
= lim ln |x − 1|
I=
= lim ln ε = −∞.
ε→0
x − 1 ε→0
x − 1 ε→0
0
0
0
f (x) (a; b]
#b
f (x) dx
a
$b
$b
f (x) dx = lim
ε→0
a+ε
a
!
f (x) dx
" # x = c # [a; b]
$b
$c
f (x) dx =
a
$b
f (x) dx +
a
$
f (x) dx.
c
% & " # '
( ) & "
$
$1
−1
dx
.
x2
* + , # " x = 0
#0
#1 dx
. ) /
,-' ' ' xdx
x
−1
0
" ) ( 0
2
$1
0
dx
= lim
ε→0
x2
$1
2
dx
1
1
= − lim |1ε = − lim 1 −
ε→0 x
ε→0
x2
ε
= +∞.
ε
(
$1
−1
dx
x2
1
$1
−1
1
dx
1
=
−
= −2.
x2
x −1
! " #
$
$ [a; +∞)
!
a
ϕ(x)
+∞
#
!"
#
f (x)
0 ϕ(x) f (x).
+∞
#
f (x) dx
"
ϕ(x) dx
"
a
ϕ(x) dx
+∞
#
+∞
#
a
f (x) dx;
a
%
# $ I =
$+∞
1
+∞
#
1
&
dx
x2 +1
dx
.
x3 + 1
' &
( )*
+∞
# dx
+ [1; +∞) x31+1 x21+1
x3 +1
,
1
) f (x) ϕ(x) [a; b)
0 ϕ(x) f (x),
% x = b &
#b
a
a
f (x) dx
a
#b
ϕ(x) dx
#b
#b
0
ϕ(x) dx
f (x) dx
$1
√
3
I=
0
dx
.
1 − x5
[0; 1)
x = 1 ! "
1
√
# [0; 1) $%
3
1−x
x = 1
& ' √
0 1 √ x5 x 1 − x5 1 − x
1
1
( 3 1 − x5 3 1 − x √
√
3
3
1−x
1−x5
1
) *
√
#
3
1−x5
1
[0; 1) √
3
1−x
$1
dx
√
+ I =
3
1−x
0
$1−ε
√
3
I = lim
ε→0
0
3
dx
3
= − lim ε2/3 − 1 = .
2 ε→0
2
1−x
) ' * '
'
+ ,, ' # ' . * & * / 0 1
% . /
u1 + u2 +, · · · , +un + · · · ,
x = 1, 2, · · · n, · · ·
f (x)
[1; +∞),
u1 = f (1), u2 = f (2), · · · , un = f (n), · · · .
$+∞
I=
f (x)dx.
1
!"
#
$%& ' ()*%*+ ' ()*%,+%
-%& ' ()*%*+ ' ()*%,+%
. ' /
1
1
1
1
+ α + α + ··· α + ··· ,
α
1
2
3
n
$+∞
dx
I =
.
xα
1
α > 1 α 1.
α > 1
α 1.
! " # $! ! %
$
& '( # ' !
!
0
I=
$+∞
e−αx dx.
0
$+∞
$b
−αx
e−αx dx =
I=
e
dx = lim
b→+∞
0
0
1
= − lim e−bx − 1 =
α b→+∞
1
bα
b→+∞ e
lim e−αb = lim
α > 0
b→+∞
b → +∞.
α > 0,
p < 0.
1
,
α
+∞,
= 0,
ebα −→ +∞
α = 0
α > 0
α1 . α 0
I=
$+∞
cos x dx.
0
+∞
I = sin x 0 = sin(+∞) − sin 0.
sin x x → +∞
! " # $
%
$1
dx
.
1 + x2
I=
−∞
I = arctg x 1−∞ = arctg 1 − arctg(−∞) = π/4 + π/2 = 3π/4.
&' " "
lim arctg x = π/2.
x→+∞
! " # $
$+∞
I=
−∞
3π/4.
arctg2 x dx
.
1 + x2
$0
I = I1 + I2 =
$0
I1 =
arctg 2 x d(arctg x) =
−∞
I2 =
−∞
arctg2 x dx
+
1 + x2
$+∞
0
arctg2 x dx
.
1 + x2
1
1
arctg 3 x 0−∞ = (0 − (−π/2)3 ) = π 3 /24,
3
3
$+∞
1
1
arctg 2 x d(arctg x) = arctg 3 x −∞
= ((π/2)3 − 0) = π 3 /24.
0
3
3
0
I = I1 + I2 = π 3 /24 + π 3 /24 = π 3 /12.
!
" #
$%&
$+∞
I=
0
dx
.
1 + x3
! '" [1; +∞) (")
" *
$%+
1
1
.
3
1+x
1 + x2
, - " . "
"/
$%0
I=
$+∞
2
e−x dx.
0
#
(x − 1)2 0 =⇒ x2 − 2x + 1 0 =⇒ −x2 −2x + 1.
! ex "
e−x e−2x+1 e−x e · e−2x .
#$% " &
2
2
I=
$+∞
e−2x dx
0
'
( '" )
!
#$*
$+∞
I=
1
arctg x dx
.
x
'
$+∞
I1 =
1
dx
.
x
+" &
$+∞
I1 =
1
dx
= ln x +∞
= ln(+∞) = +∞.
1
x
x
& & a arctg
> x1 ,
x
'
&, &" -,
#$. m
F
=
m
x2
M Ox
x
A
M x = a
$+∞
mdx
m
m
A=
− 2 = +∞
=− .
x
x a
a
a
! " ! ##$ %
&''
$1
I=
0
dx
√ .
x
( x → 0 #" % √1x
" ) # [; 1].
" " ##$ %
!!
$1
I = lim
→0
√
√
dx
√ = 2 lim x 1 = 2(1 − lim ) = 2.
→0
→0
x
* " $ !
&'+,
$2
I=
0
dx
.
x−1
( "
-! ! "
$1
I = I1 + I2 =
0
dx
+
x−1
$2
1
dx
.
x−1
I1 :
$1−
I1 = lim
→0
0
dx
= lim ln |x − 1| 1−
= lim ln = −∞.
0
→0
→0
x−1
I1
I2 !
+∞
"#$$
1
√
n n
n=1
% & ' ( "#" Un = f (n) = n√1 n
)* !
$+∞
I=
1
dx
√ =
x x
,"#-.
$+∞
1
$+∞
1
α = 3/2 +
dx
x3/2
dx
x2
/&
0! !
"#$-
"#$1
+∞
#
sin xdx.
0
$+∞
0
"#$"
$+∞
0
+∞
n=1
1
√ .
n n
"#$2
$2
√
0
dx
.
2−x
dx
.
a2 + b2 x2
xdx
.
c2 + x2
"#$3
$2
1
dx
.
x ln x
$27
−8
$1
0
dx
√
.
3
x
$+∞
−∞
dx
.
x2 + 2x + 2
arcsin x dx
√
.
1 − x2
$+∞
1
x3 + 1
dx.
x4
+∞
# √
xe−x dx.
0
$1
3
0
$1
x2 dx
.
(1 − x2 )5
dx
.
e −1
√
0
x
1
1
1
1
+
+
+ ··· ,+
+ ··· .
2 ln 2 3 ln 3 4 ln 4
n ln n
!" # $! %&
' ()
*+
, ' "
-"
#b
f (x) dx . / &
a
! f (x) 0 [a; b] 1 ' " - &
'0 F (x) / 2+ &3 "'! 4 5
$b
f (x) dx = F (b) − F (a).
a
. 1 '0 1 ' " / ! f (x)
0 /) ' )
) &
' + ' 1 / ) 1 '
' % "
!"
" ! #
$ % & f (x)
' (
) & "
! "
)
& "
% "
f (x) 0 x ∈ [a; b].
*+, -.
/! [a; b] n $
x0 = a < x1 < x2 < . . . < xn = b.
0 h h = b−a
n
! % !
y = f (x) $ A0, A1, A2 , . . . , Ai−1, Ai, . . . ,
An−1 An
y
y=f(x)
h
y0 y1
0
a
x1 x2
yn
b x
xn-1 xn
1 y = f (x) " 2 3
! n
$b
f (x)dx ≈ y1 h + y2 h + · · · yn h = h
n
yi .
i=1
a
y = f (x)
!
$b
f (x)dx ≈ y0 h + y1 h + · · · yn−1 h = h
yi .
i=0
a
"#
n−1
# !
$%
%# % %
% #&
y = f (x)
'(% )*
+ [xi−1; xi ]
(% +
& Ai−1 Ai ,
ΔSi =
yi−1 + yi
· Δxi .
2
Ai
y=f(x)
Ai-1
x
xi-1
-% (%
xi
+
(%
A0 A1 A2 · · · An .% .
+
f (x)
#b
a
f (x) dx
' %% (% ! #
*
(% #& +
#&
& )
! " # $
# h
%
Sn =
y1 + y2
yi−1 + yi
yn−1 + yn
y0 + y1
h+
h+ ...+
h+ ...+
h.
2
2
2
2
& '#
Sn = h
n−1
y0 + yn
+
yi .
2
i=1
(# & # ## & )'* + #''
)
$b
f (x) dx ≈ h
a
y0 + yn
+
yi
2
i=1
n−1
#'* + #'
,
h = b −n a ,
,-.
# ) n = 2m $ #)#
#* ' '' # ,,
Mср
y=f(x)
Mп
y=Ax2+Bx+C
Mл
yл
yср
yп
xл
x ср
xп
x
/& & # x x
" x
' f (x) # '* M M M
y = Ax2 + Bx + C.
$x
x − x
Ax2 + Bx + C dx =
(y + 4y + y ).
6
x
!
" #
Sn = (y0 + 4y1 + y2 )
b−a
b−a
+ (y1 + 4y2 + y3 )
+ ...+
6m
6m
b−a
+(ym−2 + 4y2m−1 + y2m )
=
6m
m−1
m
b−a
=
(y0 + y2m ) + 2
y2i + 4
y2i−1 .
6m
i=1
i=1
$
$b
h
f (x) dx ≈
3
y0 + y2m + 2
m−1
y2i + 4
i=1
a
h = b−a
!
2m
)
&'!,
$b
f (x) dx = lim
λ→0
(n→+∞)
a
m
y2i−1
,
%&'!&(
i=1
* + !
n
f (ξi )Δxi.
i=1
$b
f (x) dx ≈
a
n
f (ξi )Δxi ≈ h
i=1
h = Δxi = b −n a .
n
i=1
f (ξi ),
%&'!-(
ξi .
!"
y
y=f(x)
y0
h
0
x
# $ " " " %
&' ()* + , , "
$h
|y − y0 | dx.
δh =
0
- . ,
$
/
y = y0 + y (ξ)x.
-$
$h
δh =
0
|y (ξ|x dx
M1 h2
,
2
M 0 * ) y
[a; b].
1
1
= f (x)
n=
b−a
,
h
n
M1 nh2
,
δ δh · n
2
!
δ
#
b−a
M1 h.
2
$ !
&
% %
b−a
M2 h2 ,
12
b−a
M4 h4 .
δ
180
Mk = max |f (k) (x)|
δ
) *+
+
,
&
"
!
- -
!.
'
(
[a; b].
! ! % %/
0
b
$
f (x) dx − I2n |I2n − In | .
15
((
a
1 !
-0 *+
2 !
2 0 ! !
% -
! /
%
n
2n
! /
#1
(
I = 3x2dx
0
! "# $ %
& h = n1 = 0, 1 '" &( #
3
4 % .
$ !0
I=
#1
0
%
3x2 dx = x3 |10 = 1 & 0
! 0 .
. 5
I = 0,3(0,01 +
+ 0,04 + 0,09 + 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81 + 1) = 1,155
I = 0,3(0 + 0,01 + 0,04 + 0,09 + 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81) =
0+1
I = 0,3(
+ 0,01 + 0,04 + 0,09 +
= 0,855
2
+ 0,16 + 0,25 + 0,36 + 0,49 + 0,64 + 0,81 + 1) = 1,005
!
I = 0,1(0 + 1 + 2(0,04 + 0,16 + 0,36 + 0,64) + 4(0,01 + 0,09 + 0,025 +
+ 0,049 + 0,081)) = 1.
" !
!# $ %
f (x) = 3x2 f (x) = 6x f (x) = 6 f (x) = f IV (x) = 0
[0,1]
M1 = 6 M2 = 6 M4 = 0 #
& '(
)* !
#
%
#+
b−a
1
M1 h = · 6 · 0,1 = 0,3,
2
2
b−a
1
M2 h2 =
· 6 · 0,01 = 0,005,
- δ
12
12
b−a
M4 h4 = 0
( δ
180
&
.
δ
0,155 < 0,3
#
)*
0,145 < 0,3
# )*
# !# )*
) %
! #
#
%
0 !#
/
1 2
! #
# # #
1
! ! !"!#
! $ 3 %
& ' % " "
( & #
" ! )*+ " #
#1
" ! I = cos x dx ' δ 0,0001?
/
/
0
3 ) + 4 % # 567(
*) ! $ $+
M1 = M2 = M4 = 1,
# %
!*
# )* )
10−4
! # !
h 0,0002 =⇒ n
1
= 5000.
0,0002
1
0,0012 =⇒ n √
≈ 29.
0,0012
h
h
4
0,0180 =⇒ n √
4
1
≈ 2.
0,0180
! " #$% &$" "% &
'$ " ( " )
"*+ $" , " -
&$" "% " ' ./
.
"*+ 0#
$b
+∞
a
n=0
f (n) (a) (x − a)n+1 b
f (n) (a)
(x − a)n dx =
=
n!
n!
n+1
a
n=0
+∞
=
δ = 0,005
+∞
f (n) (a)
(b − a)n+1 .
(n
+
1)!
n=0
Φ(x) =
#x
0
2
e−x dx
1 0 2 ! 3 ' 4 5 1'$
" * ( * " 6 2
$x
−x2
e
0
$x
dx =
0
(1 − x2 +
x4 x6 x8
x2n
−
+
+ . . . + (−1)n
+ . . .) dx.
2!
3!
4!
n!
7 $" &" -,
x = 1 .2
x4 x6 x8
−
+
+ · · · dx =
e
dx =
1−x +
2
6
24
0
0
1
1
1 x3
1
1
x9
1 1
1 x5 1 x7
= x − 0 +
−· · · .
+
−· · · = 1− + − +
0−
3
10
42 0 216
3 10 42 216
0
$1
$1
−x2
2
0
! 2161 < 0,005 0,005
I =1−
1
1
1
78
+
−
=
.
3 10 42
105
"#$
"#%
"#&
"#'
$
$
$
$
x dx
.
3x2 − 11x + 2
x3 + 5
dx.
x3 − x2 + 4x − 4
"#" tg5 x dx.
"#(
√
√
1
arcsin x
√
dx.
x+1
√
$
$4
3x2 + 1
dx.
+x+8
x3
x
x+1
dx.
"#)
+∞
n=2
ln n
.
n(1 + ln4 n)
x = −2 y = −x y = x12 .
R = 10
$
$
(3x2 + 1)dx
d(x3 + x + 8)
=
=
3
x +x+8
x3 + x + 8
$
= d ln |x3 + x + 8| = ln |x3 + x + 8| + C.
!
"# #
# $# %&''(
$
1
u = arcsin x, du = √1−x
arcsin x
2 dx
√
√
√
dx =
=
dx
√
dv = x+1 = d(2 x + 1) → v = 2 x + 1
x+1
$ √
√
x+1
√
dx = 2 x + 1 arcsin x−
2
1−x
√
√
= 2 x + 1 arcsin x + 4 1 − x + C.
√
= 2 x + 1 arcsin x − 2
$
√
−2
dx
1−x
'
) # # *
+ ,-$
' &.' *
t = 12 (3x2 − 11x + 2) = 3x − 11
/ +
2
11
z = 3t = x − 11
/
x
=
z
+
/
dx
=
dz
"/
6
6
,-$ 3x2 − 11x + 2 = 3(x2 − 11
x
+ 23 ) =
3
121
= 3 z2 −
97
= 3 (x − 11
x z + 116 dx
)2 + 23 − 36
6
36
dz
$
$
$
xdx
1
zdz
dz
11
√
=√
+ √
=
3x2 − 11x + 2
3
z 2 − 97/36 6 3
z 2 − 97/36
11
1 2
z − 97/36 + √ ln |z + z 2 − 97/36| + C =
=√
3
6 3
1
2 11
11
2 2
11
11
2
2
=√
ln x −
+ x − x + + + C.
x − x+ +
3
3
6
6
3
3 3
3
! "#
x3 −x2 +4x−4 =
= x2 (x − 1) + 4(x − 1) = (x − 1)(x2 + 4) $
$! %
$
x3 + 5
dx =
x3 − x2 + 4x − 4
$
1+
x2 − 4x + 9
(x − 1)(x2 + 4)
dx.
x2 − 4x + 9
A
Bx + C
=
+ 2
=
(x − 1)(x2 + 4)
x−1
x +4
=
A(x2 + 4) + (x − 1)(Bx + C)
⇒
(x − 1)(x2 + 4)
⇒ A(x2 + 4) + (x − 1)(Bx + C) = x2 − 4x + 9.
x = 1
5A = 6 → A = 6/5
x = 0 4A − C = 9 → C = 4A − 9 = −21/5 & $ !
$ '(( ! x2
A + B = 1 → B = 1 − A = − 15
)$ " $
$
3
x +5
dx =
x3 − x2 + 4x − 4
=x+
$
1+
6
1 x + 21
−
5(x − 1) 5 x2 + 4
dx =
6
1
x
21
ln |x − 1| −
ln(x2 + 4) −
arctg + C.
5
10
10
2
tg x
dz
tg x = z! x = arctg z! dx = 1+z
! "
$
$
5
tg xdx =
5
z
dz =
z2 + 1
2
z4 z2
z
dz =
− +
z −z+ 2
z +1
4
2
$
3
tg4 x tg2 x 1
1
−
+ ln(tg2 x + 1) + C =
+ ln(z 2 + 1) + C =
2
4
2
2
tg4 x tg2 x
=
−
− ln | cos x| + C,
4
2
# " ! " ln(tg2 x + 1) = ln cos12 x =
= −2 ln | cos x|
$ %
#
1
1
1
1
xm (a + bxn )p = x1/2 (x1/2 + 1)− 2 → m = ; n = ; p = − , a = b = 1.
2
2
2
" %
m+1
n
=
1
+1
2
√
√
x + 1 = z 2 → x = z 2 − 1,
√
dx
√ = 2zdz, xdx = 4xzdz = 4z(z 2 − 1)2 dz.
2 x
&'" ( x
!
$4
√
√
1
x
x+1
√
=4 3
"
√
dx = 4
$3
√
2
4
2
z − 2z + 1 dz = 4
=3
1/2
√3
z5 2 3
− z + z √ =
5
3
2
√
√
√ 4 4
16 3 28 2
9
−2+1 −4 2
− +1 =
−
=
5
5 3
5
15
√
√
7 2
4
4 3−
≈ 2,903.
=
5
3
$ )
$+∞
2
$+∞
ln x
dx =
x(1 + ln4 x)
2
1
ln x
d ln x =
4
2
1 + ln x
$+∞
2
d ln2 x
=
1 + ln4 x
+∞
1 π
1
2
− arctg ln2 2 .
=
= arctg ln x
2
2 2
2
+∞
ln n
! " ! "!
#
n(1 + ln4 n)
n=2
$ ! "
%&'
! (
)*+ %, - . ( '/ ! ! #
! ( xB = −1 ! (0 1!
y=1
-x
y=
C
/x 2
y
B
A
-2
-1
$−1
−x −
−2
1
x2
dx =
x
SABC =
0
−
−1
x2 1
4 1
1
+
= − − 1 + + = 1.
2
x −2
2
2 2
%&
! 2
%' 3 " !" 0 " ! Δx
( x
2
x + y 2 = R2
ΔA = gxΔm = gxρΔV ≈ γxπy Δx = 2
y = R 2 − x2
2
=
-R
0
R
y
x
x+ Δ x
x
R
= πγx(R2 − x2 )Δx.
R
x4
−
A = πγ(R x − x )dx = πγ R
=
2
4 0
0
πγR4
π
R4 R4
−
=
= 107 ≈ 7854000
= πγ
2
4
4
4
γ = 1 3 = 103 3
$R
2
3
2x
2
.
$16
√
1
$
$
dx
.
3 cos2 x + 4 sin2 x
(x + 1)ex dx.
$
4 − 3x
dx.
2
5x + 6x + 18
$
dx
.
−1
x4
$
sin2 x
dx.
cos4 x
dx
√ .
x(1 + 4 x)3
r = 2(1 − sin ϕ).
+∞
n
.
3n2
n=1
R H !
γ
S =
ΔSi
n
ΔSi
i=1
λi ΔSi
ΔSi
λ
max λi .
ΔSi λ = i=1,2,···
,n
! " # λ → 0 "# λi → 0
$ " % "
Pi ∈ ΔSi ξi , ηi &# '( $ # f (Pi ) =
f (ξi ; ηi ) ! " f (ξi ; ηi )ΔSi ## " #
n
f (ξi ; ηi )ΔSi "# ΔSi .
i=1
f (ξi; ηi)ΔSi n
i=1
z = f (x, y)
S
* f (x, y)
S
n
n
f (ξi; ηi)ΔSi ! ! !
i=1
" ΔSi ! λ
#
n
)
+
#,
##
f (x, y)ds "#
S
##
, # S ΔSi "
Pi ∈ ΔSi -
$$
n
lim
f
(ξ
;
η
)ΔS
=
f (x, y)ds.
i i
i
n→+∞,
λ→0
i=1
& (
S
##
.#
" #, " S / 0
# " , f (x, y) / 1 ds / 0
f (x, y)ds / " ! "0
n
" # #", ,
ΔSi = S #"
i=1
lim
n→+∞,
λ→0
n
i=1
$$
ΔSi =
ds = S,
S
& )(
S
S
##
z = f (x, y) f (x, y)ds
S
! " # z =
= f (x, y) $
S
% $ & " # '( z = f (x, y)
) z = f (x, y) 0
S * !
σ
!
)
z = f (x, y) " Oxy S & +,(
- ! Pi ∈ ΔSi " ! Mi ∈ σ
Oxy
z
Mi σ
ζi
ηi
y
0
ξi
Pi S
x
. ζi = f (ξi ; ηi ) = Pi Mi f (ξi ; ηi )ΔSi = ζi ΔSi
/0 " ΔSi $ Pi Mi = ζi
n
n
1
f (ξi ; ηi )ΔSi =
ζi ΔSi /0
n=1
n=1
2 ! 3 Vn
) "
lim
n→+∞
n
$$
f (ξi ; ηi )ΔSi =
i=1
f (x, y)ds = lim Vn = V,
n→+∞
S
V
Oz S
σ ! "#$
%$ f (x, y) 0 S
&$ !! ' $ #
$" (
) # (' & )
!
• &$' ' !
*
$$
$$
kf (x, y)ds = k
f (x, y)ds,
S
•
S
+ ' !%
!% *
$$
$$
(f (x, y) + ϕ(x, y))ds =
S
•
'
$$
f (x, y)ds +
S
• ,
## )
f (x, y)ds 0
•
k = 0.
ϕ(x, y)ds.
S
$ f (x, y) 0
S
, ) $ m M $ $"$
- ) - ###$ !%
f (x, y) m f (x, y) M mS
f (x, y)ds MS
S
. ( %! m
, ) $ S = Sj
j=1
$$
f (x, y)ds =
S
m $$
j=1 S
i
f (x, y)ds.
•
f (x, y)
S S
##
P (ξ; η)
f (ξ, η) = S1
f (x, y)ds
S
f (x; y; z) S.
! "! #
!
f (x, y) 0 !
$ f (x, y) 0 S
%
%
S
y = y (x) ! & y = y (x)
x = a x = b ' ()* %
x = const
Oy a < x < b !
" $ y = y (x)
!
" !
y = y (x) + () , $
-./ -0/
y
x=const
y=yB (x)
K
B
A
S
D
C
E
a
b
y=yH(x)
x
Oy
1) 2 S
a < x < b y y = y (x) y = y(x)
OY x = const ! ! "
Oy
A D B C
y = y (x) y = y(x)
f (x, y) 0 S
f (x, y) S ! "
S σ
#$
z
z=f(x,y)
σ A’
K’
D’
E’
C’
B’
D
a
y
A
K
E
S
b
C
x
B
x =
% & ' x = ()*+, "
EE K K -. #$ -
#/
y #(x)
01 & Q(x) =
f (x, y)dy & x
y (x)
2 ! ABCDAB C D - "
1' 34# 5
$b
V =
$b
Q(x) dx =
a
a
⎛
⎜
⎝
y$ (x)
⎞
⎟
f (x, y)dy ⎠ dx.
y (x)
z
z=f(x,y)
E’
K’
y=yB (x)
Q(x)
y=yH (x)
0
E
y
K
x =
$b
y$ (x)
dx
V =
f (x, y)dy.
y (x)
a
f (x, y) 0 :
$$
f (x, y)ds.
V =
S
V
y x
$$
$b
f (x, y)ds =
y$ (x)
dx
a
S
f (x, y)dy.
!
y (x)
" # $
# $ % & '(
$ x = )*+,- y = )*+,- & S
& $ ./!
" 0 1 ! 2
$$
$b
f (x, y)ds =
S
$d
dx
a
f (x, y)dy.
c
/!
y
d
S
c
0
a
dxdy
= ds
x
b
f (x, y)
(x,y)
x = const F (x, y) ∂F∂y
= f (x, y)
y
x
!
"#
$b
$(x) (x)
$b
y
dx
f (x, y)dy = F (x, y) dx =
y
y (x)
a
a
y
"
$b
(F (x, y (x)) − F (x, y (x))) dx.
=
a
$ % & '
(
x & )
%
) S !(
) !
S
) (
% % Oy %
!
Si ) )
* %
+
S ' a x c
y = y2 (x)
y = y1 (x) c x b
y
y=yB (x)
y=yB (x)
2
1
S1 S2
S3
y=yH (x)
y=yH (x)
2
1
a
0
c d
b
x
a x d y = y 1 (x)
d x b y = y 2 (x) S
S1 , S2 , S3
$$
$$
f (x, y)ds =
S
$$
f (x, y)ds +
S1
$c
y$1 (x)
dx
=
a
y1 (x)
$$
f (x, y)ds +
f (x, y)ds =
S2
$d
f (x, y)dy +
S3
y$1 (x)
dx
c
y2 (x)
$b
f (x, y)dy +
y$2 (x)
dx
f (x, y)dy.
y2 (x)
d
##
S
O(0; 0) A(1; 1) B(1; 0)
(x + y)ds
S
!
" y = 0 x = 1 y = x # x = const 0 < x < 1
$ % &
'% ()*
% + y = y (x) = 0
'% (,* % !
+ y = y (x) = x -
S ' '
x " . / 001
y
A(1;1)
x=const
y=x
O(0;0)
y=0
B(1;0) x
$$
$1
(x + y)ds =
0
S
$1
=
0
$x
dx
0
3
(x + x /2)dx =
2
2
2
$1
(x + y)dy =
x
(xy + y 2 /2) dx =
0
0
$1
x2 dx =
0
1
3 x3
1
= .
2 3 0 2
# dx # (x2 + 3y2)dy.
2
0
1
0
! "#
#1
(x2 + 3y2)dy x
0
y $ % & '% x
$2
$1
dx
0
(x2 + 3y 2 )dy =
0
$2
=
1
(x2 y + y 3 ) dx =
0
0
2
14
x3
8
+ x = + 2 = .
3
3
3
0
(x2 + 1)dx =
0
$2
$2
$x
dx
1
x2 dy
.
y2
1/x
! x = "#$%& x2 ' ' ( (
( ! ! ! '
$2
x2 dx
1
$x
dy
=
y2
1/x
$2
=
$2
x2 −
1
(x3 − x)dx =
1
x
$2
1
1
2
dx =
dx
=
x
x
−
y 1/x
x
1
2
9
x4 x2
1 1
−
=4−2− + = .
4
2 1
4 2
4
' ) * '
! S (' +
)
#3
1
dx
x+9
#
f (x, y)dy.
x2
,
x+9
#
x2
f (x, y)dy !
! x = const -#. ! / y = x2
01 ! S ( +
! -2. ! y = x + 9 01 S 3+
4' ! S x !0 ! !' x = 1
x = 3 5 ! !
x=1
y
10
y=x+9
B
C
x=const
A
2
1
D
0
1
x=3
y=x
x
3
y = x2 y = x + 9 x = 1 x = 3
S
y = x+9
x=1
x=3
! "# $ %! &# '
y
y= 25-x
2
y=- 25-x
+ ,
#3
−4
√
dx
25−x
# 2
√
− 25−x2
x
x=3
x=-4
x=const
( ! # ) %!*#
f (x; y)dy.
2
y = x2
√
25−x
# 2
√
f (x; y)dy
− 25−x
x = const √
y = − 25 − x2 → x2 + y2 = 25!
x "
√S #! $
% & y = 25 − x2 → x2 + y2 = 25!
" S ' ( )*
S x * x = −4 x = 3' + ! $
x2 +y2 = 25 *$
x = −4 x = 3' , # #
S ' -.
/ x2 + y2 = 25 * x = −4
x = 3 * A(−4; 3), B(3; 4), C(3; −4) D(−4; −3)'
2
$4
01'0
$y
dy
2
01'-
0
$4
$2
dx
3
01'2
$2
1
x2
y3
dx
+ y2
dy
(x + y)2
$2x
dx (2x−y)dy
301'.4 301'54 *!
" # # & '
1
01'.
01'5
√
$2
$ 4x
dx
0
$1
0
x
√
2x−x2
f (x, y)dy
√
2
$3−y
dy
f (x, y)dx
y/2
$
" # $
##
%&'
(x + y)ds S
S
O(0; 0), A(1; 1), B(0; 2)
!
y B(0;2)
x=const
y=2-x
A(1;1)
y=x
x
O(0;0)
( ) * +$ ,
x = 0, y = x, y = 2 − x -.
! " / ) %''
2−x
##
#1
#
#1
# $" (x+y)ds = dx (x+y)dy = (xy+
+y
2
/2)|2−x
x dx
− x2 −
x2
)dx
2
=
#1
0
0
S
2
2
(x(2−x)+ (2−x)
−x2 − x2 )dx
2
#1
= 2 (1 − x2 )dx = 2(x −
0
x3 1
)|
3 0
#1
0
x
2
2
= (2x−x +2−2x+ x2 −
0
= 2(1 − 13 ) = 43 .
##
S
O(0; 0) A(1; 1) B(2; 0)
(x + y)ds
S
x = 0 y = x y = 2 − x
y
x=const x=const
A(1;1)
y=2-x
y=x
O(0;0)
C(1;0)
B(2;0)
x
OY ## OAC
ACB !
(x + y)ds =
=
##
(x + y)ds +
##
S
(x + y)ds
"
OAC ACB #$
2−x
##
#1 #x
#2
#
(x + y)ds = dx (x + y)dy + dx (x + y)dy =
OAC
= (xy +
0
=
0
S
#1
+
ACB
y2 x
)| dx
2 0
#1
#2
+ (xy +
1
(2−x)2
)dx
2
=
1
2
−2+
+4−
4
3
3
2
0
2
#2
y 2 2−x
)| dx
2 0
x dx + (2 −
1
6
= 43 .
1
x2
)dx
2
0
#1
1
2
= (x +
0
=
3 x3 1
|
2 3 0
x2
)dx
2
#2
0
+ (x(2 − x) +
+ (2x −
1
x3 2
)|
6 1
=
S x = x (y)
x = x(y) y = c y = d
y
A
d
B
x=x (y)
x=xnp(y)
y=const
c
D
C
x
Ox
y = const Ox
! !
"
#
x = x (y) ! $ !
" # x = x(y) %
& $ ' ()* (+*
,- S
c < y < d x (x = x(y) x = x(y))
OX y = const
Ox
. A B ' D C !
$ x = x (y) x = x(y)
/ $
x ' 0
0 , 1
!$!2 # ! 3
y4
c < x < d
$$
$d
f (x, y)ds =
S
x$ (y)
dy
c
f (x, y)dx.
x (y)
,-
$$
$d
f (x, y)ds =
$b
dy
c
S
f (x, y)dx.
a
! !" " ! #
$" % & ' (
f (x, y) = ϕ(x)ψ(y) ! & !
! )
!*") +
$$
$b
f (x, y)dxdy =
$d
ϕ(x)dx
a
S
ψ(y)dy.
,
c
' " ! & ! f (x, y) y = -./01
y)
= f (x, y) (
$2 Φ(x, y) ! ∂Φ(x,
∂x
x ! #
% +
$d
x$ (y)
dy
c
$d
f (x, y)dx =
x (y)
c
x
Φ(x, y)
(y)
x (y)
dy =
$d
(Φ(x (y), y) − Φ(x(y), y)) dy,
=
c
! ! 3 ! !*#
y
,
4 3 + 5 ! y = -./01 0 < y < 1
2 6.7 #
x = x(y) = y 2 687
x = x (y) = 1 9 & ) S & y
" : % ;
y
A(1;1)
x=y
y=const
B(1;0) x
O(0;0)
$$
$1
(x + y)ds =
$1
dy
0
S
$1
=
0
$1
(x + y)dx =
0
y
1
3
+ y − y 2 dy =
2
2
y y2 y3
+
−
2
2
2
x2
1
+ yx dy =
2
y
1
1 1 1 1
= + − = .
2 2 2
2
0
y = 0 < y < 1 !
"# $ % & ' ( ) x = x (y) = 0 #
&# # & 0 < y < 1 ' *+ & '
& x = x(y) = y 1 < y < 2 , x = x(y) = 2 − y
& S & -
'
# OAC
##
## CAB .& /01
##
2# - (x + y)ds = (x + y)ds + (x + y)ds.
S
OAC
CAB
&& 3 ' '$ ' 4
# OAC CAB && & & /0 - '4
"
$$
$1
(x + y)ds =
S
$y
dy
0
$2
(x + y)dx +
0
1
$2−y
4
dy (x + y)dx = .
3
0
y B(0;2)
x=2-y
y=const
A(1;1)
C(0;1)
y=const
x=y
x
O(0;0)
##
S
2−y
#1
#
(x + y)ds = dy (x + y)dx = 43 .
0
y
y
A(1;1)
y=const
x=2-y
x=y
O(0;0)
C(1;0)
B(2;0)
x
!! " # $
$% $ $ &'
% $% $ $%
$( $ % $% $% $(
$ $%
f (x, y)
S
a x b, c y d !"# $
S x = %&'() y = %&'() *
+ [a; b]
k
m
k [c; d] m , b − a = Δxl c − d = Δyj
j=1
l=1
Δxl = xl − xl−1 Δyj = yj − yj−1
y
ym =d
yj
yj-1
y0 =c
x0=a
xl-1 xl
x k =b x
+
n -
S = (b − a)(c − d) n = km
ΔSi = Δxl · Δyj
f (x, y) (ξl; ηj )
. !"
f (ξl; ηj )ΔxlΔyj .
+ / j 0 *
1 !"# m−
m
j=1
f (ξl ; ηl )Δxl Δyj ,
y
l
0 Δyj → 0
Δxl →
lim
k
Δxl →0
Δyj →0
=
#b #d
a
l=1
m
f (ξl ; ηj )Δxl Δyj
j=1
= lim
Δxl →0
Δyj →0
k
m
l=1
j=1
f (ξl ; ηj )Δyj
x
Δxl =
##
#b #d
f (x, y)dy dx = dx f (x, y)dy =
f (x, y)ds.
c
a
c
S
ABCD
oy ! " #
y
y=yB (x)
A
B
D
y=y H (x) C
yj
yj-1
0
x0=a
x k =b x
xl-1 xl
Oy
$
%
&
, - % y = y (x)
y = y (x) $
Δxl Δyj , &
% , , x = '()*+
,
# .
/(0 . /10
y y = y (x) y = y (x)
% 2
%
l Δxl , 3
4!5# 6
3 47!#
Ox "!#
x = '()*+ y = '()*+ %
! "
#
$3
1
$x+9
dx
f (x, y)dy.
x2
$ % & '( !
( )* ** +
( ,
x ( (
y = 9 y = 10 - ./
01 ( ** 2 3 (,
456./7.01/7081
$$
$$
$$
$$
f (x, y)ds =
S
f (x, y)ds +
DEC
f (x, y)ds +
EAKC
f (x, y)ds.
ABK
$ &
2 '( 6./ x = 1, x = √y,
x,
9 - **2
: , ;( y = ? 0
∂x
∂y
%
δ !
" δ
P dx + Qdy =
C
∂Q
∂x
## ∂Q
∂x
δ
−
∂P
∂y
## ∂Q
δ
−
∂P
∂y
∂x
> 0
dδ > 0
−
∂P
∂y
%
dδ " C #
δ
P dx + Qdy > 0
C
$ %
" "
D
∂P
∂Q
=
.
∂y
∂x
&'(
$
ydx + xdy
AB
) * + , "
#
$
ydx − xdy.
AB
ydx+xdy :
P = y Q = x
AB
∂P
∂y
= 1
" "
% %
&'- &'.
#
/ "
ydx − xdy +
P = y Q = −x ∂P
= 1 ∂Q
= −1
∂y
∂x
∂Q
∂x
=1⇒
∂P
∂y
=
∂Q
∂x
AB
∂Q
∂P
=
"
⇒
∂y
∂x
"
%
&'- &'.
P dx + Qdy
P = P (x; y) Q = Q(x; y)
D 0
$
U = U (x; y) %
&'1
!
"
P dx + Qdy
D
U = U (x; y)
D
∂P
∂Q
=
.
∂y
∂x
P dx+Qdy
U = U (x; y)
∂U
∂U
dx +
dy = P dx + Qdy.
dU =
∂x
∂y
! " ∂U
= P ∂U
= Q # "" "
∂x
∂y
∂ U
$ y " x % ∂x∂y
= ∂P
& ∂ U = ∂Q
'
∂y ∂y∂x
∂x
∂Q
∂P
∂x " " (" ()( $ *
∂y
" U (x; y) " " ' *
* % ∂P
= ∂Q
∂y
∂x
+ ! U = U (x; y)
2
2
" # $ P dx + Qdy
% & #
#
, % P dx+Qdy *" " " (
)- % A B *" $ % A B
(x;y)
(x;y)
#
#
* % x0 y0 x y '.
dU (x; y) =
P (ξ; η)dξ +
(x ;y )
(x ;y )
+ Q(ξ; η)dη "
AB
0
0
0
0
(x;y)
$
U (x; y) =
P (ξ; η)dξ + Q(ξ; η)dη + C.
(x0 ;y0 )
/0
1 ( . % x y *"
% ." ( * %
" ξ η 2%" % ( U (x; y) (( % )
C = U (x0; y0)
3 ." " /0 " " )
()- % A(x0; y0) B(x; y) 4
* ." ( ")-
ABC *" ( AB CD (" ()-(
η = y0 = ξ = x =
dη = 0 dξ = 0
$x
U (x; y) =
$y
P (ξ; y0 )dξ +
x0
Q(x; η)dη + C.
y0
y
y
B
y0
C
A
x0
x
x
U = U (x; y)
P (x; y) Q(x; y) !
"# ! $ % &' x0 = y0 = 0
$x
U (x; y) =
$y
P (ξ; 0)dξ +
0
Q(x; η)dη + C.
(
0
º
ydx + xdy
! " ydx + xdy ##$
% & & # ' ()*
% $ % y "
$x
U (x; y) =
$y
0dξ +
0
xdη = xy + C.
0
+ dU , dU = d(xy+C) = ydx+xdy
1
x
y
dx + y2 − yx2 dy
+
1
y
P = x1 + y1 Q = y2 − yx2
1 ∂Q
1
∂P
∂P
=
−
=
−
= ∂Q
∂y
y 2 ∂x
y2
∂y
∂x
!"# $# % &'
% ( ) **# *+# $ $ $
x = 0 y = 0 )$, P (x; y) Q(x; y)
) ** x0 = y0 = 1
$x
$y
1
2
x
+ 1 dξ +
− 2 dη =
U (x; y) =
ξ
η η
1
1
y
x
=
= (ln ξ + ξ)|x1 + 2 ln η +
η 1
x
x
= ln x + x − ln 1 − 1 + 2 ln y + − 2 ln 1 − x = ln x + 2 ln y + + C.
y
y
1 1
2
x
∂U
∂U
= + ./0#1
= − 2 = Q(x, y)
$ -
∂x
x y
∂y
y y
' % ! ' $
-' ' ! %3 -
#
" !
AB
*
2xydx − x2 dy O(0; 0) A(2; 1) !
O
A # $$%&'
• - OaA
2
1
4 - OaA y = x ⇒
2
1
dy = dx
2
2
$
$2
1
4
1
2
2xydx − x dy =
x2 − x2 dx = x3 = .
2
6 0 3
OaA
0
y
A(2;1)
C(0;1)
c
y=Const
a
b
O(0;0)
• ObA
Oy
x
dy = dx
2
$
2xydx − x2 dy =
$2
0
ObA
x
B(2;0)
x 3 x3
−
2
2
• OcA Ox
ObA y =
x2
⇒
4
dx = 0.
OcA y =
x
⇒
2
dx
dy = √
2 2x
$
2
$2
2xydx − x dy =
OcA
0
√ $2
√
1 √
3 2
x 2x − x 2x dx =
x3/2 dx =
4
4
0
2
√
12
3 2 2 5/2
· x = .
=
4
5
5
0
• OBA
OBA
OB BA OB − y = 0 ⇒ dy = 0 BA − x = 2dx = 0
$
$1
2
2xydx − x dy = 0 +
(−4)dy = −4.
0
OBA
• OCA
OCA
OC CA OC − x = 0 ⇒ dx = 0 CA − y = 1 ⇒ dy = 0
$
$2
2
2xydx − x dy =
OCA
#
2
2xdx + 0 = x2 0 = 4.
0
2xydx + x2 dy
OA
•
! OaA
$
2
$2
2xydx + x dy =
0
OaA
2
$2
1 2
x3
3
2
= 4.
x dx =
x + x dx =
2
2
2 0
2
0
•
" ObA
$
2
$2
2xdx + x dy =
0
ObA
x3 x3
+
2
2
$2
dx =
0
2
x4
x dx =
= 4.
4 0
3
•
" OcA
$
2
$2
2xdx + x dy =
0
OcA
=
√
1 √
x 2x + x 2x dx =
4
√ $2
√
5 2
5 2 2 5/2
· x = 4.
x3/2 dx =
4
4
5
0
•
$
OBA
2xdx + x2 dy = 0 +
$
$1
4dy = 4.
0
OBA
•
OCA
2xdx + x2 dy =
$2
2
2xdx + 0 = x2 0 = 4.
0
OCA
!
L
• L " OBAaO
P = 2xy Q = −x2
#
∂Q ∂P
−
= −2x − 2x = −4x,
∂x
∂y
&
2xydx − x dy = −4
L
$1
= −4
0
$1
$$
2
xdS = −4
S
2
$1
x2
dy = −2 (4 − 4y 2 )dy = 8
2 2y
0
$2
dy
0
xdx =
2y
1
y3
16
− y = − .
3
3
0
$ % &
% x y = '()*+ %
y = 2x x = 2
P = 2xy Q = x2 ⇒
= 2x
2xydx + x2 dy
L
! " # $ " %
"& $ L "# O(0; 0) A(2; 1) #
'
(
∂Q
#∂x
∂P
∂y
2xydx + x2dy
) # " !%
**+ , ) #
# # # # ! $ - ,%
, * $
.
$y
U (x; y) = 0 +
x2 dη = x2 y + C.
o
/ # $ 0
∂U
∂U
= 2xy 1
= x2 .
∂x
∂y
#
2
2
(x − 2xy)dx + (2xy + y )dy
AB y = x2
A(1; 1) B(2; 4)
#
(2a − y)dx + xdy C
C
x = a(t − sin t) y = a(1 − cos t) 0 t 2π
2 !" " #
%
2(x2 + y 2 )dx + (x + y)2 dy
C
C "
$ " % A(1; 1) B(2; 2) C(1; 3) &
" "
"
" " ABCA
.3 '
x = a cos3 t y = a sin3 t
AB
(3x2 − 2xy + y 2 )dx − (x2 − 2xy + 3y 2 )dy
ex−y ((1 + x + y)dx + (1 − x − y)dy)
dy
dx
+
x+y x+y
!
" # $ % & #
' ( ( #
)! ! !
&
* (
! " #
#
+
,# ( D Oxyz
#
F ! Fx = P = P (x; y; z)- Fy = Q = Q(x; y; z)
Fz = R = R(x; y; z)- !. # x, y, z
D !
/
F = F (x; y; z) = P (x; y; z)i + Q(x; y; z)j + R(x; y; z)k
0 /1
" # $ %
/2 & L !
F = (P ; Q; R)
' ! "
#
# ! ! C
/3 (
) !" " C " "
z
Δσi
ni
Pi
σ
ζi
ηi
x
y
ξi
σ
n D
n σ = Δσi Pi ξi ηi ζi
i=1
! Δσi "
# ni = n(ξi; ηiζi) # σ
$ n σ #
%! # & % Oz '& #
& $# Fi = F (ξi; ηi; ζi)
ni Fi # Δσi #
Pi ( n Fi · ni · Δσi
n)% & % *n = Fi · ni · Δσi
+,- n
n
i=1
n = Fi ·ni ·Δσi
Δσi
i=1
n → +∞ F
σ
$$
$$
σ = F · ndσ = F (x; y; z) · n(x; y; z)dσ.
+,.
σ
σ
/ n #
0 % σ
σ
$$
=
F · ndσ.
σ
!" #" $ D %
$&" $ W = W (x; y; z)
D " ' $ # #"
F = W (x; y; z)
()# # # ) " # # %
* +" &
" #" ) "
' , " # # %
& " #
+ " ' # # %
# - ' '
& # # #
(
) # )
$ σ # # + " * #%
$& γ
- ) # # " ) $ W
## " $& %
* Δσ ./0
W
Δσ1
n
h
ϕ
Δσ
Δσ
1 + # + " %
* Δσ " # * Δσ1 W
" ) 2 ΔQ"
Δσ
Δσ W
γ h ΔQ = γhΔσ
n ! Δσ ϕ ! "
W ΔQ = γW cos ϕ · Δσ = γ(W n)Δσ
# $ %&' ()
* * σ
$$
W (x; y; z) · n(x; y; z)dσ.
Q=γ
%&' +)
σ
&' &
F
= (P (x; y; z); Q(x; y; z); R(x; y; z)) M(x; y; z) ∈ ΔV
F σ
ΔV M
ΔV → 0
div
$$
1
F ndσ
ΔV →0 ΔV
div F = lim
σ
∂Q ∂R
∂P
+
+
∂x
∂y
∂z
$$
$$$
=
F · ndσ =
div F dV.
div F =
"σ
σ
!
#
V
$ % &'
( )* ) + )* )
#
! "# $$$%% & ' (
div F = 0
##
##
F ndσ +
σ1
σ1
σ2
F ndσ = 0.
σ2
n
σ2
F
n1
F
n
σ1
n1 = −n
##
F n1 dσ =
σ1
##
n
σ
F ndσ
σ2
div F = 0
! "
div F = 0
#
B
%
L
$
D #
n
A
$
n
F (ξi ; ηi ; ζi )Δli =
i=1
=
n
P (ξi ; ηi ; ζi )Δxi + Q(ξi ; ηi ; ζi )Δyi + R(ξi ; ηi ; ζi )Δzi .
i=1
#
n
i=1
" &
n
F (ξi ; ηi ; ζi )Δli
!
! !
AB
n → +∞
$
$
F (x; y; z)dl =
L
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz
L
$
$
F (x; y; z)dl =
AB
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz.
AB
z = z(t)
L
$
x = x(t) y = y(t)
P dx + Qdy + Rdz =
AB
$tB
=
(P (x(t); y(t); z(t))x(t) + Q(x(t); y(t); z(t))y (t)+
tA
+R(x(t); y(t); z(t))z (t)) dt,
tA
#
tB
!
"
A
B
%&%
F = F (x; y; z) = (Fx(x; y; z); Fy (x; y; z); Fz (x; y; z))
( ) *
A
t
$
$
B
-
+
$
F (x; y; z)dl =
A=
AB
'
( (,
'
L
Fx (x; y; z)dx + Fy (x; y; z)dy + Fz (x; y; z)dz.
AB
!
L " " F
L
&
&
F dl =
P (x; y; z)dx + Q(x; y; z)dy + R(x; y; z)dz.
L
L
/ 0
L
1
. (" " 1%
)
"- "
1
.
- 3
( (&
4&%
%
'
(" 2 '
("
F = P (x; y; z)i + Q(x; y; z)j + R(x; y; z)k
rot F
∂R ∂Q
∂P
∂Q ∂P
∂R
−
−
−
rot F =
i+
j+
k =
∂y
∂z
∂z
∂x
∂x
∂y
i j k
∂ ∂ ∂
= ∂x ∂y ∂z .
P Q R
! " #
$" % & rot F ' !
$% ! ( ! (
! ( $" ∂x∂ ) ∂y∂ ) ∂z∂ & P * Q* R
+ $ ∂P
) ∂P ) ∂P ) ∂Q
∂x ∂y ∂z ∂x
* % ! & rot F = ∂R
− ∂Q
; ∂P − ∂R
; ∂Q − ∂P
∂y
∂z ∂z
∂x ∂x
∂y
%
,"$! rot F & "& F dl " "
L
* #$ $ !
- P = P (x; y; z)!
Q = Q(x; y; z)! R = R(x; y; z)! "#
F =
F (x; y; z)! $ σ !
L! F
L % & σ'
&
$$
F dl =
L
n rot F dσ.
.
σ
/ . $" ! $
" ' % % *
& "&" $ L " "
% & % ( *
. rot V $ + !*
L 0 " $" * + !
! "
P= P (x; y)# Q = Q(x; y)
k & $
R = 0 $ % rot F = ∂Q
− ∂P
∂x
∂y
L ' " σ ( ( Oxy n = k )
* + rot F n (
F = (P (x; y; z); Q(x; y; z); R(x; y; z))
rot F = 0
, U = U (x; y; z)
F = =F (x; y; z)
F = F (x; y; z)
- . /0
% ! F = F (x; y; z) =
"
P (x; y; z)i+Q(x; y; z)j +R(x; y; z)k
rot F =
∂R ∂Q ∂P
∂R ∂Q ∂P
−
;
−
;
−
∂y
∂z ∂z
∂x ∂x
∂y
= 0.
# $ $
$
∂R ∂Q
−
= 0;
∂y
∂z
∂P
∂R
−
= 0;
∂z
∂x
∂Q ∂P
−
= 0.
∂x
∂y
" '1. " " ( F = F (x; y; z) )
( 2 U = U (x; y; z) . " $ ( ./
, / $ . %3
F = grad U =
.
P =
∂U
,
∂x
∂U
∂U
∂U
i+
j+
k
∂x
∂y
∂z
Q=
∂U
,
∂y
R=
∂U
.
∂z
ξ η ζ
(x; y; z) C = U (x0 ; y0 ; z0 )
z
B(x;y;z)
A(x 0;y0 ;z 0)
C(x;y0 ;z0)
D(x;y;z 0)
y
x
L
ACDB AC CD
DB
!""#
η = y0 ζ = z0 $ ξ = x ζ = z0 $ ξ = x η = y dη = dζ = 0$ dξ = dζ = 0$
dξ = dη = 0 % & ' ()!*#
$x
U (x; y; z) =
$y
P (ξ; y0 ; z0 )dξ +
x0
$z
Q(x; η; z0 )dη +
y0
R(x; y; ζ)dζ + C.
z0
()"+#
x0 =
()! P Q R
y0 = z0 = 0
O(0; 0; 0)
()" ! " # $
# %& &' "' U = U (x; y) $
# ( " # %& )' "' U =
= U (x; y; z) *
" + + $
# " #
& , # - #
F = kr = kxi +
R
+kyj + kzk
x2 + y 2 + z 2 = R2
n = R1 R
F = kR F n = kR · R1 R = Rk R2 = kR
F = kr
x2 + y 2 + z 2 = R2
$$
$$
σ =
F ndσ = kR
dσ = kR4πR2 = 4πkR3 .
σ
σ
! " # " "
P = kx$ Q = ky $ R = kz % &
div F = ∂P
+ ∂Q
+ ∂R
= ∂kx
+ ∂ky
+ ∂kz
= 3k #
∂x
∂y
∂z
∂x
∂y
∂z
' " &( )
σ =
$$$
$$$
div F dV = 3k
V
4
dV = 3k · πR3 = 4πkR3 .
3
V
*
F = (2x + z)i + (y + 2z)j + (z − y)k
x = 0 y = 0 z = 0 x − 2y + 2z − 6 = 0
+ , - . / "
0 1 2*3 ' ! " F
# " " , P = 2x+z $ Q = y +2z $ R = z −y
div F =
∂(2x + z) ∂(y + 2z) ∂(z − y)
+
+
=2+1+1=4
∂x
∂y
∂z
)
σ =
$$$
$$$
div F dV = 4
V
dV =
V
= 4 · 61 OA · BO · OC = 4 16 · 6 · 3 · 3 = 36.
z
3
y
0
C
A
6 x
y=const
x=6+2y
x=6-2z
z=const
0
A
6 x
y = 0
•
3B
z = 0
ABC : x − 2y + 2z − 6 = 0
N = Ai + Bj + Ck = i − 2j + 2k ⇒ n =
N
=
N
i − 2j + 2k
2
2
1
=√
= i − j + k.
3
3
3
1+4+4
F n
F n = 13 (2x + z − 2y − 4z + 2z − 2y)
z = 3 − x2 + y
F n = 56 x − 52 y − 1
! "#$ %& dσ = 1 + z 2x + z2y dxdy =
= 1 + 14 + 1dxdy = 32 dxdy
'ABC = 23
$$
5
5
x − y − 1 dxdy =
6
3
AOB
3
=
2
3
=
2
=
3
2
$0
−3
#0
−3
$0
6+2y
$
dy
−3
0
5
5
x − y − 1 dx =
6
3
5
5
2
(6 + 2y) − (6 + 2y) − (6 + 2y) dy =
12
3
9 − 2y − 53 y 2 dy =
3
2
0
9y − y 2 − 53 y 3 −3 = 31,5.
OABC σ = BOC + AOC + AOB +
+ ABC = −4,5 + 18 − 9 + 31,5 = 36.
! " " #
$
"
% & '()#
*+#,
•
z = x2 + y2
x2 + y2 = 4 !"
F = xy 2 i +
yz
j
2
+ x2 zk
$ - .
/ -
z = f (x; y) 0 %'(#1()
−fx (x; y)i − fy (x; y)j + k
−2xi − 2yj + k
n=
.
=
1 + 4x2 + 4y 2
1 + f 2x (x; y) + f 2y (x; y)
/ n Ox
#
−2x2 y 2 − y 2 z + x2 z
F n =
#
1 + 4x2 + 4y 2
2 "
=
$$
σ
3
2 2
2
2
−2x y − y z + x z
dσ.
1 + 4x2 + 4y 2
0 %,4#55)
dσ =
1 + f 2x + f 2y ds =
1 + 4x2 + 4y 2 ds.
/ z = x2 + y2 0
=
$$
(−2x2 y 2 + (x2 + y 2 )(x2 − y 2 ))dσ,
σxy
σxy 6 R
#
= 2
Oxy &
=
$$
(−2r4 cos2 ϕ sin2 ϕ + r4 (cos2 ϕ − sin2 ϕ))rdrdϕ =
σxy
$2
sin2 2ϕ
=
+ cos 2ϕ dϕ r5 dr =
−
2
0
0
2π 6 2
2
ϕ sin 2ϕ sin 2ϕ r
16π
=
+
+
6 = − 3 .
4
16
2
0
$2π
0
F
ACBA
! P = 2x + z" Q = y + 2z" R = z − y
" " # $ % &'(
∂R ∂Q
−
i+
∂y
∂z
rot F =
∂P
∂R
−
∂z
∂x
j+
∂Q ∂P
−
∂x
∂y
k=
= (−1 − 2)i + (1 − 0)j + (0 − 0)k = −3i + j.
n
) *
$$
$$
n rot F dσ =
σ
5
2
(−1 − )dσ = −
3
3
ACB
=−
rot F ABC
5 SABC
5
=−
3 cos γ
3
$$
5
dσ = − dσACB =
3
ACB
1
OA
2
· OB
536·3
=−
= −22,5.
2/3
32 2
+ , " # $ % & ( -$-
$$ ACBA
&
$$
n rot F dσ = −22,5.
F dl =
ACBA
ACB
%
. -$-
F dl /
ACBA
P dx + Qdy + Rdz , AC 0 CB BA
• AC y = 0 #z = 3 − x2
dy = 0 dz = − 12 dx
(2x + y)dx + (y + 2z)dy + (z − y)dz =
#0 7
6
4
x+
3
2
dx =
7
8
AB
0
x2 + 32 x 6 = −40,5.
dx = 0 dz = dy
• CB x = 0 # z = 3 + y
(2x + y)dx + (y + 2z)dy + (z − y)dz =
−3
#
CB
(9 + 3y) dy = 9y +
0
3y 2
2
−3
= −13,5.
0
dz = 0 dx = 2dy
• BA z = 0 # x = 6 + 2y
(2x + y)dx + (y + 2z)dy + (z − y)dz =
BA
2 0
(24 + 9y) dy = 24y + 9y2 = 31,5.
−3
−3
%
F dl = −40,5 − 13,5 + 31,5 = −22,5
#0
ACBA
F = (2xy + z 2 )i + (x2 + z)j + (y + 2xz)k
!
" P = 2xy + z 2 # Q = x2 + z # R = y + 2xz $
% & '()
∂R ∂Q
∂P
∂R
−
= 1 − 1 = 0;
−
= 2z − 2z = 0;
∂y
∂z
∂z
∂x
∂Q ∂P
−
= 2x − 2x = 0,
∂x
∂y
% rot F = 0 *+
, &('-.) / x0 = y0 = z0 = 0 %
0
+ , P (x; y; z)# Q(x; y; z) R(x; y; z)
$x
$y
$z
2
u(x; y; z) = 0dx + x dy + (y + 2xz)dz = x2 y + yz + xz 2 + C.
0
0
0
∂u
∂u
∂u
= 2xy + z 2 # Q =
= x2 + z # R =
= y + 2xz.
$ P =
∂x
∂y
∂z
F = xi+yj+zk
x = −a x = a
y = −a y = a z = −a z = a
F = xyi + yzj + xzk
! " x2 + y2 + z2 = 1
#"
F = yzi + zxj + xyk $
!"#
$ %
&
' & %
( )
)
%
L
$" L = AB " % ' ' *
'
+ S
Z , -./
z
y
zn
B
zi
A
z0
z i-1 ζ
i
S
x
n !
" !
" " L# # zA = z0 zB = zn $!
% L " # &" '
( ") zi−1 zi * " ξi n
f (ξi )(zi − zi−1 ) = f (ξi )Δzi
" ) " "" f (ξi)Δzi
L
i=1
+, f (z)
n z L
f (ξi)Δzi
i=1
! "
L "
ξi #
$
f (z)dz = lim
n→+∞
L
n
+,
f (ξi )Δzi .
i=1
$ % & )
+, " ) L " !
# "
" # z
* -" f (z) ./
# z = x + iy# f (z) = u + iv# ) u = u(x; y)# v = v(x; y) 0 !
( z f (z) ) # " # i2 = −1#
$" $
$
$
f (z)dz = (u + iv)(dx + idy) = udx − vdy + i vdx + udy. +/
L
L
L
#
L
#
1 # Re f (z)dz = u(x; y)dx−v(x; y)dy
L
L
#
#
Im f (z)dz = v(x; y)dx + u(x; y)dy )
L
L
L ) !
2 3 ) # "
2 )
L
#
z = 1 + 2i
Imzdz
L
y
z
2
z
1
ϕ
Β
x
1
1
0
Α
1
y = 2x dy = 2dx
Imz = Im(x + iy) = y
$
$1
$
Imzdz =
0z
y(dx + idy) =
0z
$1
2xdx + i
0
4xdx =
0
1
1
= x2 0 + i · 2x2 0 = 1 + 2i.
# |z|dz
L
L
z = cos ϕ y = sin ϕ (0 ϕ π)
z = cos ϕ + i sin ϕ dz = (− sin ϕ + i cos ϕ)dϕ
|z| =
x2 + y 2 + cos2 ϕ + sin2 ϕ = 1.
$
$π
(− sin ϕ + i cos ϕ)dϕ = cos ϕ|π0 + i sin ϕ|π0 = −1 − 1 = −2.
|z|dz =
AB
0
!" #$%
#& '
&
dz
z
L
y
z
1
|z|=1
0
1
x
( ) * + z , z = |z|eiϕ = eiϕ
!" ''% dz = ieiϕ dϕ (0 ϕ 2π) -
&
dz
=
z
|z|=1
$2π
0
ieiϕ dϕ
= iϕ|2π
0 = 2πi.
eiϕ
#& .
! "# f (z) = u(x; y) + iv(x; y)% " $
S % f (z)dz & ' ! " !!
L
$
Re
#
L
zA zB
#
f (z)dz =
L
L
f (z)dz
LAB
# #
f (z)dz = vdx + udy
udx − vdy Im
L
L
! "#$ $ !
$
P dx + Qdy
LAB
% ! L %
#$ & #' " "#$ $%
∂P
=
() ∂Q
∂x
∂y
!
$
udx − vdy : P = u, Q = −v →
∂v ∂P
∂u
∂Q
=− ;
=
,
∂x
∂x ∂y
∂y
L
!
$
vdx + udy : P = v, Q = u →
∂u ∂P
∂v
∂Q
=
;
=
.
∂x
∂x ∂y
∂y
L
* & $ + f (z) = u+iv "
∂v ∂u
∂v
=
, $% & - . /0 ∂u
1 = − ∂x
∂x
∂y ∂y
() $%# $ 2 #
" ! f (z)dz %
L
! %
$ & 3
#
! f (z)dz # 4 "
L
&
f (z)dz = 0.
5
L
* % & !
" + ! $' # ' + 5 %
# % % n
#
n
n
f (z) 3x
S L0 L1 L2
!"!#
y
Z
L2
B
L1
B
B
H
A
B
C
B
H
H
D
H
L0
S
x
$% A ∈ L0 % B ∈ L1 C ∈ L1
% D ∈ L2 & % ' '
&
L = A Lo A B L1 C D L2 D C L1 B A ,
f (z) ( ) % * + *+
, $
&
$
$
$
$
$
f (z)dz =
+
+
+
+
+
L
A L0 A
A B
$
B L1 C
$
+
+
D C
C D
D L2 D
$
+
C L1 B
= 0.
B A
#
#
# # #
#
%
= −
= −
+
= − f (z)dz - &
A B
B A C#D
L1
% D C B L1 #C C L%1 B
= f (z)dz
= f (z)dz
D L2 D
L2
A L0 A
L0
&
&
&
f (z)dz =
L0
n
f (z)dz +
L1
L2
&
f (z)dz =
n &
k=1 L
L0
f (z)dz.
!
f (z)dz
k
"#"
$
f (z) S
L0
Lk
% &' "( )
* +
&
%
!" # (z − a)mdz L $
L
" m $
, - . % a L &(
'( (z − a)%m (( ( L " m
(z − a)m dz = 0
L
Z
y
|z−a|=δ
a
S
L
x
δ
L
a
a
L
m ≥ 0 %
(z − a)mdz = 0
L
m < −1
a
!
" L # # δ
a
# L $ %&'( ) δ
C : |z − a| = δ z = a + δeiϕ
S
* (z − a)m 2x +
L C
, -.'
&
&
m
m
(z − a) dz = (z − a) dz =
L
C
&
=
(z−a)=δ
dz
=
(z − a)k
$2π
0
m = −1
k = −m > 0
z − a = δeiϕ
dz = iδeiϕ dϕ
iδeiϕ dϕ
= iδ 1−k
δ k eikϕ
$2π
0
=
dϕ
=
ei(k−1)ϕ
=
2π
e−i(k−1)ϕ
iδ 1−k
(1 − k)i 0
= 0,
ez T = 2πi
m = −1 $-.&(
&
|z−a|=δ
/
dz
=
z−a
$2π
0
iδeiϕ dϕ
= iϕ|2π
0 = 2πi.
δeiϕ
&
(z − a) dz =
m
2πi, m = −1,
0, m = −1.
$-.-(
L
f (z)
a 0 +
L
! "
a
%
f (z)
1
2πi
f (z)dz
L
Resf (z) =
1
2πi
&
f (z)dz.
L
f (z) a
f (z) = · · · +
C−1
C−2
C−1
+
+ ··· +
+
(z − a)n
(z − a)2 z − a
+C0 + C1 (z − a) + · · · + Cn (z − a)n + . . .
!" #$ % & 0 < |z − a| < R' %
%' (
#% R
a' $ a !" L "
0 < |z−a| < R ! ) ( ( *
&
&
&
&
+∞
+∞
dz
dz
+
f (z)dz =
C−m
+
C
C
(z − a)k dz.
−1
k
(z − a)m
z − a k=0
m=2
L
L
L
L
+
& ) (
% dz ) # (
( # ' z−a
= 2πi' # ' ,
L
%
" ' f (z)dz = C−1 2πi' #
L
#
C−1
* $
$ & $ a *,
f (z)
-
Resf (a) = C−1 .
. &#% f (z)' ,
) $ & $ ,
/$ ) $ ' ( n,) ( 0
) " n ' / & $ $
0 & + ) ' '
1 a ( n,) ( f (z)'
* $ 2
C−n
C−n+1
C−1
+
+
+ ··· +
Ck (z − a)k .
(z − a)n (z − a)n−1
z − a k=0
+∞
f (z) =
(z − a)n
(z − a) f (z) = C−n + C−n+1 (z − a) + · · · + C−1 (z − a)n−1 +
n
+
+∞
Ck (z − a)k+n .
k=0
n − 1
n−1
d
((z − a)n f (z)) = C−1 (n − 1)! + C0 n(n − 1) . . .
dz n−1
. . . 2(z − a) + C1 (n + 1)n . . . 3(z − a)2 + . . .
z → a
dn−1
((z − a)n f (z)) = C−1 (n − 1)! = (n − 1)!Resf (a).
z→a dz n−1
f (z)
lim
n
Resf (a) =
dn−1
1
lim n−1 ((z − a)n f (z)) .
(n − 1)! z→a dz
!"#$%
5
!"#!
f (z) = (z +z 1)4
1
& ' ( ) f (z)
= (z+1)
z = −1
z
z
*+ f (z) = (z+1)
,
- *+ # - *.%# !"#$%
4
5
5
4
z 5
d3
1
lim
=
Res
(z + 1)4 z=−1 3! z→−1 dz 3
=
(z + 1)4
z5
(z + 1)4
=
1
1
5·4·3
lim (z 5 )III =
lim 5 · 4 · 3z 2 =
= 10.
z→−1
z→−1
3!
3!
1·2·3
/ a
0 0! = 1 f 0 (z) = f (z) !"#$%
n = 1
Resf (a) = lim (z − a)f (z).
!"#"%
z→a
!"#1
z2
= 2
z −1
f (z) =
1
f (z)
= z z−1 = (z−1)(z+1)
z
a1 = 1 a2 = −1 f (z) = z z−1
!!"
2
2
2
2
2
z2
1
z 2
z2
= lim
= ,
= lim(z − 1) 2
2
z − 1 z=1 z→1
z − 1 z→1 z + 1
2
z2
z 2
1
z2
Res 2
= lim
=− .
= lim (z + 1) 2
z→−1
z −1
z − 1 z→−1 z − 1
2
Res
z=−1
ϕ(z)
# f (z) f (z) = ψ(z)
$ ϕ(z) ψ(z) %
a & ϕ(a) = 0 a
& ' ψ(z) ψ(a) = 0 ψ (a) = 0
' !("
Resf (a) = lim (z − a)
z→a
ϕ(z)
= ϕ(a) lim
z→a
ψ(z)
) * '
ϕ(a) = 0 ψ(a) = 0 ψ (a) = 0
Res
1
ψ(z)−ψ(a)
z−a
a
ϕ(z)
ϕ(a)
.
=
ψ(z) z=a ψ (a)
!-
=
ϕ(a)
.
ψ (a)
f (z) =
ϕ(z)
ψ(z)
!+,"
f (z) = z z−1
a1 = 1
a2 = −1 . ϕ(z) = z2 * / .
!+,"
2
2
Res
z 2
z 2
z 2
1
z 2
1
;
Res
=
=
=
=− .
2
2
z − 1 z=1
2z z=1 2
z − 1 z=−1
2z z=−1
2
# a f (z)
f (z)
C−1
!( f (z) = e1/z
z = 0
) z = 0 / 0 * 0 0
f (z) = e1/z 1 &
234
e1/z = 1 +
1
1
1
+
+ ··· +
+...
z 2!z 2
n!z n
C−1 = 1
Res e1/z z=0 = 1
f (z)
S
L !
ak ∈ S k = 1, 2 . . . , n
L "# $ "
f (z) % # 2πi
&
f (z)dz = 2πi
n
Resf (ak ).
k=1
L
ak
! γk " ! # $ %$
& " ! # #
y
z
a2
a1
an
γ2
γ1
γn
S
L
x
' #! ( # ( L γ1 γ2 . . . γn
) * ! f (z) ( & +
&
f (z)dz =
& ,
L
n &
k=1 γ
f (z)dz,
k
&
f (z)dz = 2πiResf (ak ),
γk
&
z4
|z|=3/2
ez dz
+ 3z 2 − 4
z
e dz
! f (z) = z4 +3z
"
2 −4
4
2
$ z + 3z − 4 = 0 % z 4 + 3z 2 − 4
% z 4 + 3z 2 − 4 = (z − 1)(z + 1)(z − 2i)(z + 2i) $
# a1 = 1 a2 = −1 a3 = 2i a4 = −2i !!&! '
' & ( % |z| = 3/2 $ !!
a1 a2 a3 a4 % ) *
#
= 0
"
$
y
Z
2
|z|=1
|z|=3/2
1
-1
1
0
2 x
-2
+
"
&
|z|=3/2
ez
+
Res 4
z + 3z 2 − 4 z=1
ez
.
+ Res 4
z + 3z 2 − 4 z=−1
ez dz
= 2πi
z 4 + 3z 2 − 4
z = 1
! !
Res
z4
ez
ea
,
= 3
2
+ 3z − 4 z=a 4a + 6a
z = −1
e
Res 4
z + 3z 2 − 4
a=1
z
z=1
e
;
10
a = −1
e
Res 4
z + 3z 2 − 4
z
=−
z=−1
&
|z|=3/2
ez dz
= 2πi
4
z + 3z 2 − 4
e
1
−
10 10e
=
e−1
1
=−
.
10
10e
πi 2
(e − 1).
5e
!" #$
$2π
0
dϕ
(a > 1)
a + cos ϕ
% & ' (
e = z dz = ieiϕ dϕ = izdϕ cos ϕ =
z 2 +1
= 2z )
ϕ
0 2π z
=
|z| = 1 *
iϕ
eiϕ +e−iϕ
2
$2π
0
dϕ
=
a + cos ϕ
=
&
|z|=1
dz
2
=
z 2 +1
i
iz(a + 2z )
&
dz
=
z 2 + 2az + 1
|z|=1
z 2 + 2az + 1√= 0,
1
=
z1,2 = −a ± a2 − 1 , Res z2 +2az+1
z1
z1 ∈ |z| < 1,
z2 ∈ |z| > 1,
= 2√a12 −1
1
2z+2a z1
=
2
1
2π
= 2πi · √
=√
.
i 2 a2 − 1
a2 − 1
+ & ,
-
,
N .
#/!0 N → +∞
y
z
N
aj
a2
a1
N
x1
an-1
an
0 xk xm-1 x m N x
x2
f (z)
aj
f (z)
Imz ≥ 0
j = 1, 2, . . . n Imaj > 0
$+∞
n
f (x)dx = 2πi
Resf (aj ).
−∞
!
j=1
$+∞
−∞
x2 + 1
dx"
x4 + 1
!" f (z) = zz +1
#! $ %
+1
& '( ! ) * !
+ ,,- !" ,,
2
4
√
a1,2 =
a21,2 + 1
z 2 + 1
2
(i ± 1); Res 4
=
=
2
z + 1 a1,2
4a31,2
1 2
(i ± 2i + 1) + 1
2
4
√ (i3 ± 3i2 + 3i ± 1)
2 2
√
√
√
21±i i±1
2(i ± i2 ± 1 + i)
2i
±i + 1
= 4
·
=
=−
.
=
2
√ (i ∓ 1)
4 i∓1 i±1
4(i − 1)
4
2
=
$+∞
−∞
√
√
√
√
x2 + 1
2
2
2
dx = 2πi −
i−
i = 2πi −
i = π 2.
4
x +1
4
4
2
!"
# $ % & '#% (
f (z) = g(z)eiαz α > 0 g(z) → 0
|z| → +∞
Imz ≥ 0 aj j = 1, 2, . . . , n (Imaj > 0)
xk k = 1, . . . , m x !"#
n
$+∞
m
1
f (x)dx = 2πi
Resf (aj ) +
Resf (xk ) .
2 k=1
j=1
)
−∞
$+∞
$
−∞
sin x
dx
x
$+∞
−∞
cos x
dx
x
* + , - . #/ (
' 0) 1 & - )))
ix
e = cos x + i sin x
$+∞
−∞
eix
dx =
x
$+∞
−∞
cos x
dx + i
x
$+∞
−∞
sin x
dx.
x
2 '#
# #
→ 0 |z| → +∞ 4
# 3 α =
e
5 z = 0 Res z z=0 = e1 z=0 = e0 = 1
iz
f (z) = ez
1 g(z) = 1z
iz
$+∞
−∞
cos x
dx + i
x
$+∞
−∞
sin x
1
eiz
dx = 2πi Res
= πi.
x
2
z z=0
4 #
$+∞
−∞
iz
5 5
cos x
dx = 0,
x
6
$+∞
−∞
sin x
dx = π.
x
$+∞
−∞
sin x
dx
x4 − 1
$+∞
−∞
cos x
dx.
x4 − 1
$+∞ ix
$+∞
$+∞
e
cos x
sin x
dx =
dx + i
dx.
x4 − 1
x4 − 1
x4 − 1
−∞
−∞
−∞
iz
f (z) = ze4 −1 !"
# α = 1" g(z) = z41−1 → 0 z → +∞ " z 4 − 1 =
= (z − 1)(z + 1)(z 2 + 1) $ %
x : a1 = 1"
a2 = −1 &# ' % a3 = i $ '
( $ $ )* + ,!
cos 1 + i sin 1
eiaj
ei
eiz
;
Res 4
= 3 ; Resf (1) = =
z −1
4a
4
4
z=aj
j
− cos 1 + i sin 1
e−1
i
e−i
=
; Resf (i) =
= .
Resf (−1) = −
4
4
−4i
4e
- + ,!
$+∞
$+∞
cos x
sin x
dx + i
dx =
x4 − 1
x4 − 1
−∞
−∞
1
= 2πi Resf (i) + (Resf (1) + Resf (−1)) =
2
i
cos 1 + i sin 1 − cos 1 + i sin 1
π
= 2πi
+
= − − 2π sin 1.
4e
2
2e
.
$+∞
$+∞
1
cos x
sin x
dx = −π
+ 2 sin 1 ,
dx = 0.
x4 − 1
2e
x4 − 1
−∞
−∞
$1
$3−x
dx
0
√
2y(x+2y 2 )dy
1−x2
z = x2 x = y2 x = 1 z = 0
&
(3xy + x2 )dx + 8x2 dy
ABCA
A(0; 1)! B(2; 2)! C(0; 3)
" #
$" %
!
F = (x + y; x + y + z; 2z − y) & #
O(0; 0; 0)! A(−3; 0; 0)! B(0; 2; 0)! C(0; 0; 3)
$" ' %
(
F = (ex sin y; ex cos y; 1)
" # $
$1
$3−x
dx
√
1−x2
0
$1
=
0
2
$1
2y(x + 2y )dy =
3−x
(y 2 x + y 4 )√1−x2 dx =
0
(3 − x)2 x + (3 − x)4 − x(1 − x2 ) − (1 − x2 )2 dx =
$1
=
(9 − 6x + x2 )(9 − 5x + x2 ) − x + x3 − 1 + 2x2 − x4 dx =
0
$1
=
(81 − 54x + 9x2 − 45x + 30x2 − 5x3 + 9x2 − 6x3 + x4 − x + x3 − 1+
0
2
4
$1
+2x − x )dx =
(80 − 100x + 50x2 − 10x3 )dx =
0
=
80x − 50x2 +
1
265
50 4 10 4
50 5
x − x = 30 +
− =
.
3
4
3
2
6
0
x = 0 x = 1 y = 3 − x
y 2 = 1 − x2 ! ABCE
"#$
y
3
2
1
C
x=3-y
K
E
B
D
A
0
1
x= 1-y 2
2
3
x
ADE EDKB KBC
ABCE
$1
$3−x
dx
√
0
$2
+
2y(x + 2y )dy =
$1
2y(x + 2y 2 )dx +
0
$3
=
+
0
$1
$1
2y(x + 2y 2 )dx =
dy
√
0
1−y 2
2
3−y
yx + 4y 3 x 0 dy +
$2
2
$1
$3−y
dy
2y(x + 2y 2 )dx+
2
1−x2
dy
1
$3
2
1
yx2 + 4y 3 x 0 dy+
1
1
yx2 + 4y 3 x √1−y2 =
$3
y(9 − 6y + y 2 ) + 4y 3 (3 − y) dy+
2
0
$2
+
(y + 4y 3 )dy +
1
$1
(y + 4y 3 − y(1 − y 2 ) − 4y 3
1 − y 2 )dy =
0
$3
2
3
4
$2
(9y − 6y + 13y − 4y )dy +
=
2
(y + 4y 3 )dy+
1
3
9y 2
13y 4 4y 5
− 2y 3 +
−
+
2
4
5 2
1
2
5 4 4 (1 − y 2 )5 4 (1 − y 2 )3
y2
4
+y +
y −
+
=
2
4
5
3
1
$1
5y 3 − 4y 3 1 − y 2 dy =
+
0
+
0
1053 972
81
− 54 +
−
− 18 + 16 − 52+
=
2
4
5
128
1
5 4 4
265
+
+ 2 + 16 − − 1 + + − =
.
5
2
4 5 3
6
OABCD
OCD z = x2 OAD OBC
x = y 2 x = 1 z = 0
!"# $%
z
y
b)
D(1;1;−1)
x=Const
a)
C(1;1;1)
y= x
B
OX
1
y
A(1;−1;0)
0
x
S A
B(1;1;0)
y=− x
x
& ' ( ) *+
'
!"# .%
$$
$$
V =
zds =
S
x2 dxdy = 2
$1
, (
√
$x
$1
√x
x2 dx dy = 2 x2 y 0 dx =
0
S
$1
=2
0
0
0
1
√
2
4
x2 xdx = 2 · x7/2 = .
7
7
0
/ y = 0 0 ) , x 1 2
3!"%
x =
7
4
$$$
xdv =
V
7
4
-
√
$1
$x
xdx
0
$x2
dy
dz =
√
− x
0
0
− x
7
4
$1
0
3!%
x2
√
$ x
xdx
z dy =
√
− x
0
√ x
$1
$x
$1
$1
√
7
7
7
=
x3 dx
dy =
x3 y
dx =
2x3 xdx =
4
4
4
√
√
√
0
− x
0
z
1
7 2 9
7
· x2 = ;
2 9
9
0
=
z =
7
4
$$$
7
4
zdv =
V
=
7
8
$1
√
$1
$x
dx
√
x4 dx
$x
dy =
√
− x
0
=
$$
=
$x2
dy
√
− x
0
7
8
$1
0
zdz =
0
7
4
$x
dx
√
− x
0
√x
7
x4 y −√x dx =
4
$1
x 2
z 2
dy =
2 0
√
x4 xdx =
0
1
7
7 2 11
x2 = .
4 11
22
0
&
∂Q ∂P
−
∂x
∂y
√
$1
P dx + Qdy =
L
dxdy
y
3
C
x=Const
S
z
y=3-
2 S
B
1A
y=1+
2
3
1
x
2
C
n
A
-3
1
x
2
B
2
0
x
y
x
P = 3xy + x2
&
2
Q = 8x2
∂Q
∂x
−
ABCA
13xdxdy = 13
0
3− 12 x
$
xdx
0
S
= 13
= 16x − 3x = 13x
$2
$$
2
(3xy + x )dx + 8x dy =
$2
∂P
∂y
dy =
1+ 12 x
$2
x
x
dx = 13 (2x − x2 )dx =
x 3− −1−
2
2
0
2
52
x
8
= .
x2 −
=
13
4
−
3 0
3
3
3
= 13
AB : y = 1 + 12 x dy = 12 dx
BC : y = 3 −
dy = − 12 dx CA : x = 0 dx = 0
!" #$
1
x
2
&
$2
x
3x 1 +
+ x2 + 4x2 dx+
(3xy + x )dx + 8x dy =
2
2
2
0
ABCA
$0
+
2
$0
$2
x
3x 3 −
+ x2 − 4x2 dx + 0dy = (−6x + 11x2 )dx =
2
3
=
0
2
11
52
−3x2 + x3 = .
3
3
0
%& '
( ) *#+, - $#. &
V = 61 OA · OB · OC =
1
6
·2·3·3=3
/ * 01 0
F $ - σ
2σ
$$
$$$
div F = ∂(x+y) + ∂(x+y+z) +
∂x
∂y
= F ndσ =
divF dV = ∂(2z−y)
∂z = 1 + 1 + 2 = 4
σ
V
$$$
dV = 4V = 12.
=4
V
=
z
x=const
3 C
z=x+3
A
-3
0 x
OAC
n=
N
N
= − √217 ; √317 ; √217
3
2
1
2
F n = − √ (x + y) + √ (x + y + z) + √ (2z − y) = √ (x − y + 7z) =
17
17
17
17
ABC
=
=
z = 3 + x − 32 y
23
21
1
1
=√
x − y + 21 + 7x − y = √ (8x − y + 21),
2
2
17
17
$$
1
23
√
8x − y + 21 dσ =
2
17
ABC
= dS = dxdy = dσ · cos γ =
$2
$0
21
23
4x − y +
dxdy = dy
4
2
$$
=
0
OBA
$2
=
2x2 −
−2
=
0
$2
−
=
0
3
y−3
2
=
23
21
4x − y +
dx =
4
2
3
y−3
2
0
21
23
y + x
dy =
4
2
3
y−3
2
0
$2
√2 dσ
17
2
23
+ y
4
3
21
y−3 −
2
2
3
y−3
2
dy =
69y 2 69
63
63
−9y 2
+ 18y − 18 +
− y− y+
dy =
2
8
4
4
2
$2
=
0
33 2
27
y − 15y +
dy =
8
2
2
11 3 15 2 27
y − y + y =
8
2
2
0
= 11 − 30 + 27 = 8,
$$
F ndσ = 2 + 2 + 0 + 8 = 12.
σ
F = F (x; y; z)
rot F = 0 !" #
$
k
i
j
∂1 ∂ex cos y
∂
∂
∂
=
−
i+
rot F = ∂x
∂y
∂z
∂y
∂z
ex sin y ex cos y 1
∂ex cos y ∂ex sin y
∂1 ∂ex sin y
+
j+
−
k=
+ −
∂x
∂z
∂x
∂y
= 0 · i + 0 · j + (ex cos y − ex cos y)k = 0.
% & ' $ ()" & * +
, , x0 = y0 = z0 = 0 $
$x
$y
$z
x
V (x; y; z) = 0dξ + e cos ηdη + dζ + C =
0
x
sin η|y0
x
0
ζ|z0
0
x
+
+ C = e sin y + z + C.
=e
∂V
∂(e sin y + z + C)
=
= ex sin y = Fx
% & *
∂x
∂x
∂V
∂(ex sin y + z + C)
=
= ex cos y = Fy ;
∂y
∂y
∂V
∂(ex sin y + z + C)
=
= 1 = Fz .
∂z
∂z
$1
1−x
$ 2
dx
−1
√
− 1−x2
xydy
x2 + y2 = z + 1 z = 3
% 2xydx + (x2 − 3xy)dy A(1; 1)
ABCA
B(2; 2) C(3; 1) ! " ABC
#$ %
F = (2z; x−y+z; 3y+2z)
& " O(0; 0; 0) A(−4/3; 0; 0)
B(0; 2; 0) C(0; 0; 4) #$ '
%
(
F = (2xz + y 2 ; 2xy + z 2 ; 2yz + x2 )
Oxy
x + y = 4 Oxy
x + y = 3 (x + 1) +
+(y − 1) = 4 (x + 3) + y = 1
+ y = 1 z x + 9y + 18y + 9
z < x + 2x + y /4 − y/2 + 5/5
2x − z − 1 > 0 y + 2x + 1 > 0
(x + 3) + y = c
+
=1
c > 18 ! "
(x − 1) + (y − 1) = 4
! " + y = 1 # ! "
x − 1 = (y + 1) #
! " y − 1 = x
+ z − = 0.
! # Oz.
$ ! # Oy. !
# Oy. ! # %
Oz. & #
! # Ox.
(x + 1) + +
= 1 '( (x − 1) +
+(y + 1) + z = 3 ! # − + z = 1
! # (y − 1) − + z = 1 $
! # +z − = −1 $ !
= −1
# (x − 1) + y −
! #
+
+
y =
! # x =
! # z = − − (y + 1) &
# ! # y = − $ !
# + Y − = −1 λ = 3 λ = 6 λ = −2 det A = −36
det D = 216
+ + = 1 λ = 2 λ = 5
λ = 8 det A = 80 det D = 2560
+ + = 1
λ = 1 λ = 2 λ = 3 det A = 6 det D = −42.
2
2
2
2
2
2
2
2
x2
4
2
2
2
2
2
2
x2
4
2
y2
4
2
(z−1)2
1/2
y2
1/4
2
x2
2
x2
3
2
2
x2
3
(z+1)2
3
2
z2
3
x2
2
X2
2
y2
2
2
2
∂z
∂x
4
2
x2
2
Z2
3
2
9
z2
3
1
X2
16
2
Y2
32/5
y2
2
(y+1)2
4
2
x2
3
Z2
4
3
1
(y+1)2
c−18
2
2
2
2
x2
9(c−18)
2
2
x2
4
2
1
X2
7
2
(z−1)2
4
2
Y2
7/2
Z2
7/3
3
= 4x − 2y + 3;
∂z
∂y
= −6y − 2x − 5.
∂z
∂x
=
−3(2x−5y)−2(2y−3x)
;
(2x−5y)2
ln (
√
x+1)12
x
12
−
12
√
x
12
+ C. −
√
(3−4x)(9−5x)
5
+ 2021√5 ln |51 − 40x +
+ C. √16 ln |12x +
5(3 − 4x)(9 − 5x)| + C. − 3 x+1
x−2
√
2
√
7 + 8x − 11x2 + 1163
·
+5 + 2 6(6x2 + 5x + 11)| + C. − 11
11
+4
√
1 15+3x
√
+ C. arccos x√1 2 + C. − 15
+ C.
· arcsin 11x−4
x
93
3
x
1
3+4x
−3
− 5√x2 −5 + C. x
. 3x + 4 = t2 .
x
h
n−1
f (a + ih), h =
i=0
1
2
(a
2
b−a
.
n
heh
eh −1
2
eb − ea , h =
− b2 ) . π6 . e3 − e2 .
2
1 + ln 94 . π4 . πa4 .
−2.
ln
π
2
− 1.
b−a
.
n
√
1+ 5
.
2
π
.
2
2
323 .√ 38π
. 32π
. 16 . πc2 .
R2 √
a2
√
x
2
2
2π 5. 2 4a x + 1 + 4a1 ln
2ax + 4a2
x2 + 1 .
6R. a2 ϕ + 1 + ϕ2 + ln ϕ + 1 + ϕ2 .
π
.
3
gt2
.
2
√
2abπ . 2 2.
π8 . π.
2
arctg 2√tg3x + C xex + C 29
arctg 5x+3
−
45
9
1+x 1
1
ln(5x + 6x + 18) + C − 4 ln 1−x − 2 arctg x + C
3
tg3 x + C 187 6π
12 γπHR4
3
− 10
1
√
2 3
2
√
ln(25/24). 76 y =
2x − x2 ,
2
y = 4x, x = 0, x = 2. x = y /2, x = 3 − y 2 , y = 0, y = 1.
√
1− 1−y 2
√
√
#1
#
2
y = 2x − x , y = 4x, x = 0, x = 2, dy
f (x, y) dx +
√
14π/3.
+
#1
0
dy
1+
#
√
1−y 2
f (x, y) dx +
√
2# 2
1
0
dy
#2
y 2 /4
f (x, y) dx.
y 2 /4
x
=
y 2 /2,
√
1/2
#2x
#
# 2 #1
3 − y 2 , y = 0, y = 1, dx f (x, y) dy +
dx f (x, y) dy +
x =
√
√
+
#3
√
dx
0
3−x
# 2
0
1/2
0
f (x, y) dy.
0
2
π/4
#
dϕ
1/ #
cos ϕ
0
0
3π/4
#
ρf (ρ cos ϕ, ρ sin ϕ) dρ+
dϕ
dϕ
1/#
sin ϕ
0
ρf (ρ cos ϕ, ρ sin ϕ) dρ.
0
π/4
1/#
sin ϕ
π/4
π/2
#
ρf (ρ cos ϕ, ρ sin ϕ) dρ.
4 4 3
π a.
3
2
π(1−e−a ).
√
3(π/4 + 1/2. 17/24.
x = 5a/6, y = 0. 21
πa4 γ, 49
πa4 γ,
32
32
dx
1−x
#
√
# 1−x−y
R#2 −x2
#R
#H
f (x, y, z)dz.
dx
dy f (x, y, z)dz.
0
√
0
0
0
−R
− R2 −x2
√
√
√
√
5
8
15 (31 + 12 2 − 27 3. 4π 2/3. πa5 (18 3 − 976 ).
83 R3 (π − 43 ). π/10.
#1
35
πa4 γ.
16
dy
z = 4/3, x = y = 0, x = y = 0, z = 2H/3.
πH 2 (R24 − R14 )/4. 14 πH 2 (R24 − 3R14 + 2R12 R22 ), 14 πH 2 (3R24 −
−R14 − 2R12 R22 ).
40 19
−2πa2 − 43
30
ex−y (x + y) + C ln |x + y| + C
24a3
#1
−1
dx
1−x
# 2
√
− 1−x2
3π
16
3
πa2
8
x3 −x2 y−y 3 +C
u = xyz + C
√
√
2
2
#1−y
#1
#1−y
xydy =
dy
xydx + dy
xydx = 0
√
√
−1
0
x = y = 0 z =
= xy 2 + yz 2 + x2 z + C
#0
−
5
3
1−y 2
−4
−
16
9
1−y 2
U (x; y; z) =
," "
$"
," " $"
#
," " "
##
," " "
#
-("
%
- . ( "
( * .
- .
%#
- ""
( *
#
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$" #
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.
-
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&
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/ . . %%
/ %
/ *
/
&
!"
#
!" "
$ #%
!
! &
! &
! %
' &%
'
' " &
' " (
&
'
' )
&%
'
' *
&
'
' +""
#
!
"#
$ %& & ' (
% !"
$'
) *
$% )
!!
$+ "
$
#
$% ,$ -.%/+ !.%/+ 0) ,,,
.%/+ '
( () ,,!
. % ,
.+ #
1 %( *
1 %
1
1% ' ) ,
10
10 0) ,,
1 ) ,"
1 0)
1
1 0 *
1 *"
1) % 1 -,
1 #
1
#!
1 )
,2 ) (
"
1' %
,""
1' )
,!,
1' ) 3 -
4&
$3 #
4 )
5 ,
5 ) -
5 "
5 (
) ",
5 (
5( )
6 %
6 7 *,
6 . 27
6 5 -
6 0 )
,!
6 ,
8 ) ( )
*
8 !*
9'( +
,!
9 $3 2 % ,!"
9 5 ,#,
9 ,#,
:( )
:( "
:) )
; ' 2 )
##
; ' &
) #
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