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Файл:Mathematics (Математика). Учебное пособие
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221
Variant 14.
2;
xxdx
æö
ç÷
èø
;
dx
sin2;
xdx
a)
;
dx
exdx
2
;
14
dx
x
Calculate the integrals
1
2
ò
a)
Variant 15.
Calculate the integrals
ò
-+
0
2
0
2
x
13
( )
2
3
x+
3
x
b)
b)
1
ò
0
1
ò
13
0
2
x
-
x
x+
2
2
ò
1
c)
1
ò
c)
0
Topic 16. Applications of a definite integral.
Improper integrals
Variant 1.
To calculate improper integrals or to establish their divergence:
¥
dx
ò
e
а)
Variant 2.
To calculate improper integrals or to establish their divergence:
1
ò
0
а)
Variant 3.
To calculate improper integrals or to establish their divergence:
¥
ò
0
а)
;
3
ln
xx
б)
2
dxx
;
3
1
- x
б)
-
x
;
dxe
б)
1
dx
- x
xdx
2
.
2
.
4
+x
.ln xdx
ò
1
0
¥
ò
0
1
ò
0
-
222
Variant 4.
2
1
1
1
2
Calculate the improper integral or set its divergence
1
dx
;
ò
3
x
0
а)
Variant 5.
Calculate the improper integral or set its divergence:
а)
Variant 6.
Calculate the improper integral or set its divergence:
а)
Variant 7.
Calculate the improper integral or set its divergence:
а)
Variant 8.
Calculate the improper integral or set its divergence:
а)
Variant 9.
Calculate the improper integral or set its divergence:
а)
¥
ò
¥
ò
e
+¥
4
ò
¥
ò
0
xdx
;
2
4
+x
б)
dx
;
ln
xx
dx
ò
2
¥-
dx
2
-
x
dxxe
;
84
++ xx
б)
.
2
86
-- xx
;
б)
б)
б)
б)
+¥
ò
¥-
2
ò
1
ò
-
ò
¥
1
ò
0
2
dx
( )
-x
dx
( )
+x
e
dx
ln
dx
ò
3
e
dx
+ xx
1
ln
dx
.
2
.
4
1
.
3
xx
xx
.
2
.
116
++ xx
.
223
Variant 10.
-
1
-
Calculate the improper integral or set its divergence:
б)
б)
б)
¥
ò
2
1
ò
3
2
ò
3
0
3
ò
0
e
ò
1
0
ò
3-
xdx
2
( )
5
+x
dx
2
x
.
dx
-
1 x
dx
2
-
9 x
dx
2
×
ln
dx
+
3
x
2
xdx
ò
4
0
а)
Variant 11.
Calculate the improper integral or set its divergence:
¥
x
dxe
ò
¥
а)
Variant 12.
Calculate the improper integral or set its divergence:
¥
cos dxx
ò
0
а)
Variant 13.
Calculate the improper integral or set its divergence:
¥
dx
ò
1
0
а)
Variant 14.
Calculate the improper integral or set its divergence:
¥
dx
ò
0
а)
Variant 15.
Calculate the improper integral or set its divergence:
¥
dx
ò
0
а)
;
2
- x
; б)
×
; б)
1
2
+
x
; б)
+
1
x
2
x
2
x
++
22
xx
;
+
4
;
.
3
.
.
xx
.
.
224
Section 5. Integral calculation
(
)
(
)
of multi-variable functions
Topic 17. Multiple integral
Variant 1.
1. Change the order of integration:
2. Calculate the double integral:
Variant 2.
1. Change the order of integration:
2. Calculate the double integral:
33
òò
Variant 3.
1. Change the order of integration:
2. Calculate the double integral:
2
òò
=³= xyyxyDdxdyxy 4,0,:,
òò
D
Variant 4.
1. Change the order of integration:
2. Calculate the double integral:
23
òò
D
0
-
1
D
ò ò
1
0
==+
xyxyDdxdyyx ,:,
1
0
yxyDdxdyyx 0,1:,2
+-
62
x
( )
,
dyyxfdx
( )
,xxdyyxfdx
.
22
xyxyDdxdyxy
2,:,
==
.
.
òò
2
-
8
x
2
-314
0
.
3
-
8
y
( )
,
dxyxfdy
òò
--
44
y
.
.
+44
x
( )
,
dyyxfdx
òò
3
8
x
.
=-=-
.
225
Variant 5.
(
)
(
)
(
)
(
)
2
1. Change the order of integration:
2. Calculate the double integral:
òò
2
D
Variant 6.
1. Change the order of integration:
2. Calculate the double integral:
òò
D
Variant 7.
1. Change the order of integration:
2. Calculate the double integral:
òò
D
Variant 8.
1. Change the order of integration:
2. Calculate the double integral:
òò
-
dxdyx ,2
D
,: === xxyxyD
Variant 9.
1. Change the order of integration:
2. Calculate the double integral:
3
òò
D
3
8
y
( )
òò
--
6201
y
.
==+
xyxyDdxdyy 5,:,1
3
8
x
( )
òò
--
4401
x
22
+-=-=+
xyxyDdxdyyx 1,1:,
.
+62
y
1
( )
òò
3
0
8
y
.
===-
xxyxyDdxdyyx 3,,5:,1
3
-
8
1
x
( )
òò
--
0
62
x
.
,
dxyxfdy
,
dyyxfdx
.
,,
dxyxfdy
.
,
dyyxfdx
.
1
2,
.
+-
44
y
0
( )
,
dxyxfdy
òò
-
3
1
-
8
y
.
2,0,:,
===
xyxyDdxdyxy
.
226
Variant 10.
(
)
(
)
1. Change the order of integration:
2. Calculate the double integral:
3
òò
==== xyyxyD .
Variant 11.
1. Change the order of integration:
2. Calculate the double integral:
2
òò
D
Variant 12.
1. Change the order of integration:
2. Calculate the double integral:
y
òò
D
Variant 13.
1. Change the order of integration:
2. Calculate the double integral:
32
òò
D
Variant 14.
1. Change the order of integration:
2. Calculate the double integral:
òò
D
1
0
D
3,0,8,:
0
ò ò
- -
1
===
xyxyDdxdyxy 1,0,:,
-
1
ò ò
-
5
===
xyxyDdxdye 2,0,ln:,
5
ò ò
3
==+
xyxyDdxdyxy 3,:,1
-
1
ò ò
-
3
yxyxyDdxdyxy
3
-
8
x
( )
,
dyyxfdx
òò
--
44
x
+
dxdyyx ,
3
8
y
( )
44
y
.
,
dxyxfdy
.
.
2
--
6
( )
,yydxyxfdy
.
0
.
0
( )
,xxdyyxfdx
.
2
--
8
.
0
( )
,yydxyxfdy
.
2
---
4
2,0,:, =+==
.
227
Variant 15.
(
)
1. Change the order of integration:
2. Calculate the double integral:
2
7
ò ò
1
òò
D
0,1,1:
³-==+ xxyyxD
2
-
8
( )
,,yydxyxfdy
.
0
3
+
dxdyyx ,3
.
if
Topic 18. Contour and surface integrals
To prepare papers on the topic with the obligatory analysis of the solution of example.
228
Section 6. Differential equations
(
)
¢
xysin
=¢¢
(
)
=
¢
()(
)
x
5
=+¢
yyx
(
)
=+-
¢
¢
x
cos
x
(
)
=
¢
(
)
¢
Topic 19. Basic differential equations
Variant 1.
Find the general solution of the differential equation:
а)
Variant 2.
Find the general solution of the differential equation:
а)
Variant 3.
Find the general solution of the differential equation:
+
а)
Variant 4.
Find the general solution of the differential equation:
а)
Variant 5.
Find the general solution of the differential equation:
а)
Variant 6.
Find the general solution of the differential equation:
а) xy y x
Variant 7.
Find the general solution of the differential equation:
а)
22
-
=
xyyyx ln
¢
;
xy
2 2 2
- = ;
¢
xyyyx 2
;
;
yxeyx
xyyyx ln
б)
22
121 xxyyx +=-
б)
0=-+
;
;
б)
2
;
1
¢¢
y
б)
¢¢
y =
б)
+1
б)
.
4
dyyxdyydx
22 =-
¢¢
y
=
б)
xxyy
1
=
2
.
1
.
.
.
6
3
.
xx
eyye =
;
229
Variant 8.
(
)
=++
ydxxdy õ dy
3sin
yx
¢¢
=
(
)
¢
3
4
x
-
(
)
+=¢
3
cosyx
xxyyx
cos2cos
sin
+=¢
2
cosyx
(
)
¢
x
()(
)
4
x
¢¢
=
5
=+¢
yyx
sin3
yx
¢¢
=
Find the general solution of the differential equation:
а)
Variant 9.
Find the general solution of the differential equation:
xy
а)
y
Variant 10.
Find the general solution of the differential equation:
1
а)
Variant 11.
Find the general solution of the differential equation:
а)
Variant 12.
Find the general solution of the differential equation:
2
=
¢
2 2
-
x y
3
2
-
xdydxyx
;
xyyx =
02
;
б)
б)
;
xyxy sin1cos
;
-=
¢¢
y
=
б)
¢¢
=
б)
22
.
.
2
.
4
.
а)
Variant 13.
Find the general solution of the differential equation:
22
а)
-
Variant 14.
Find the general solution of the differential equation:
+
а)
Variant 15.
Find the general solution of the differential equation:
а)
=
¢
;
xyyyx 2
;
2
22
121 xxyyx +=-
;
б)
;
б)
б)
¢¢
=-
1
¢¢
y
=
2
.
.
y
б)
.
.
230
Topic 20. Liner differential equations
670
¢¢¢
--=
67;(0)1,(0)1.
--=-==
40
¢¢
+=
43;(0)2,(0)1.
+=-==-
=+¢-¢
¢
3
27
690,
¢¢¢
-+=
4130,
¢¢¢
-+=
=¢=+=+¢-¢
¢
40,
¢¢¢
-=
20,
¢¢¢
-=
464, (0)1, (0)0.
-=+==
40
¢¢
+=
45;(0)1,(0)1.
+=-==-
Find a particular solution of the differential equation:
Variant 1.
yyy
а)
¢¢¢¢
yyyxxyy
б)
Variant 2.
а)
yy
¢¢¢
yyxyy
б)
Variant 3.
а) .096
¢¢- ¢+ = - + = ¢ =y y y x x y6 9 3 0
б)
Variant 4.
yyy
а)
¢
¢¢
б) .271)0( ,34)0( ,396
-
Variant 5.
yyy
а)
.
2
yyy
2
2
2
, () , () . y
4
=+-=+
¢
yyxxyyy
1
0
=
б) .0)0( ,1)0( ,526134
Variant 6.
yy
а)
¢¢
-
б)
Variant 7.
yy
а)
¢¢¢¢
yyxyy
б)
Variant 8.
yy
а)
¢¢¢
yyxyy
б)
2
¢
2
.
2
¢
=+=
yyxyy
yyxyyy
.3)0( ,2)0( ,164
=
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