Интегрирование функций. Лекция № 10

1.

Лекция № 10
Интегрирование
функций.

2.

x2
c
1. xdx
2
3
2
3
x
x
x
2
2
2
(
x
2
x
)
dx
x
dx
2
xdx
2
c
x
c
2.
3
2
3
3. (7 sin x
cos x
1
1
)dx 7 sin xdx cos xdx 7 cos x sin x c
4
4
4

3.

4.
1
1 4
1 y5
y5
(2 x 3)5
(2 x 3) dx y 2 dy 2 y dy 2 5 c 10 c 10 c
y 2x 3
4
4
dy (2 x 3)1 dx
dy 2dx
dx
1
dy
2
5. sin x(3 7 x)dx sin y ( 17 dy) 17 sin ydy 17 cos y c 17 cos(3 7 x) c
y 3 7x
dy (3 7 x)1 dx
dy 7dx
dx
1
dy
7

4.

6.
7.
8.
2 x 2 x3
(2 x )dx 2 dx x dx ln x 3 c
x
2
x
2
5dx
dx
5
cos2 x cos2 x 5tgx c
5dx
5
x
cos 2 (2 )
3
x
y 2
3
x
dy (2 )1 dx
3
1
dy dx
3
dx 3dy
3dy
x
15
tgy
c
15
ty
(
2
) c
cos 2 y
3

5.

9.
2
x
7
2
dx 2 y 2dy 2 2 y dy 2
7
2
2 2
c
ln 2
ln 2
y
x
2
8
c
2
c
ln 2
x
dx
2
x
dy (7 )1 dx
2
1
dy dx
2
dx 2dy
y 7
10.
x
y
y
y
x
l
dx
l
(
dy
)
l
dy
l
c
l
c
y x
dy ( x)1 dx
dy dx
dx dy
x
2

6.

3dx
dx
1
x
x
3
3
arctg
c
arctg
c
11.
2
2
2
9 x
3 x
3
3
3
2
2
2 xdx 2 xdx 2
12.
1
1
2 2
x
2
2
x 2 22 12 4 1 3
1
1
2
2
13. sin xdx cos x
cos
0
0
2
cos 0 0 1 1

7.

14.
15.
2
3
32
3 2
9
1
81 1
80
3
dx
ln 3 2 ln 3 ln 3 ln 3 9 ln 3 9 ln 3 9 ln 3
2
2
x
x
2
1
1
4 1
1
1
1 y
3
(
2
x
3
)
dx
y
dy
y
dy
1
1 2
2 1
2 4
3
y 2x 3
dy (2 x 3) dx
dy 2dx
1
dy
2
yв 4 3 1
dx
yн 2 3 1
3
1
4 1
y
8
1
14 ( 1) 4 1 1
0
8
8
8 8

8.

0
16.
0
0
0
x3
x2
2
2
3
2 0
(
3
x
2
x
)
dx
3
x
dx
2
xdx
(
3
2
)
(
x
x
) (03 02 ) (( 1)3 ( 1) 2 )
1
1
1
1
3
2 1
( 1 1) 0
17.
6
x
3
2
2
2
2y
22
12
4
1
3
9
y
y
2
dx
2
(
3
dy
)
3
2
dy
3
3
(
)
3
(
)
3
3
1
1
ln 2 1
ln 2 ln 2
ln 2 ln 2
ln 2 ln 2
y
x
3
x
dy ( ) dx
3
1
dy dx
3
dx 3dy
6
2
3
3
ун 1
3
yв

9.

3
3
3
18.
x3
x2
x3
33
1
1
2
2
2
2
(
x
2
x
1
)
dx
(
2
x
)
(
x
x
)
(
3
3
)
(
1
1
)
3
2
1
3
2
3
3
3
3
3
1
1
19.
3
6
6
6
1
1 y
1 y
1 6 4
2x
y
l
l
(
dy
)
l
dy
l
(l l )
2
4 2
24
2 4 2
y 2x
dy (2 x) dx
dy 2dx
1
dy
2
yв 2 3 6
dx
yн 2 2 4

10.

3
dx
20. 5 2 x
1
7
dy
dy 7 dy
7 y 1 y 1 y ln y
y 2 x
dy (2 x) dx
dy dx
dx dy
y в 2 3 1
yн 2 5 7
7
1
ln 7 ln 1 ln 7

11.

4
4
21. 2 cos x dx 2 cos y 4dy 8 cos ydy 8 sin y
4
0
y
x
4
x
dy ( ) dx
4
1
dy dx
4
dx 4dy
yв
4
yн 0
0
0
4
0
8(sin
4
sin 0) 8(
2
1) 4 2 8
2