Basic Integration Problems
I. Find the following integrals.
2
1. (5 8 5)x x dx
32
2. ( 6 9 4 3)x x x dx
3
2
3. ( 2 3)x x dx
23
8 5 6
4. dx
x
xx
1
5. ( )
3
x dx
x
5
3
3
4
6. (12 9 )x x dx
2
2
4
7.
x
dx
x
1
8. dx
xx
2
9. (1 3 )t t dt
2
3
11. y y dy
12. d
13. 7sin( )x dx
14. 5cos( )d
15. 9sin(3 )x dx
16. 12cos(4 )d
17. 7cos(5 )x dx
7
19. 4
x
e dx
4
20. 9
x
e dx
21. 5cos x dx
6
22. 13
t
e dt
II. Evaluate the following definite integrals.
4
2
1
1. (5 8 5)x x dx
3
2
9
1
2. ( 2 3)x x dx
9
4
1
3. ( )
3
x dx
x
4
3
1
5
4. dx
x
2
2
1
5. (1 3 )t t dt
1
22
2
6. (2 1)t dt
Solutions
I. Find the following integrals.
3
22
5
1. (5 8 5) 4 5
3
x
x x dx x x C
4
3 2 3 2
3
2. ( 6 9 4 3) 3 2 3
2
x
x x x dx x x x C
3
2
5
2
2
2
3. ( 2 3) 3
5
x
x x dx x x C
23
23
12
2
8 5 6 8
4. 5 6
5 6 5 3
8 ( ) 8 ( )
12
dx x x dx
xx
xx
xx
Ln x Ln x C
x
x
11
22
31
31
22
22
11
5. ( )
3
3
1 2 2
31
3 3 3
22
x dx x x dx
x
xx
x x C
8
7
5
3
3
4
3
4
48 27
6. (12 9 )
78
xx
x x dx c
2
2
2
44
7. 1 4
x
dx x dx x C
x
x
3
2
12
8. dx x dx C
x x x
34
2 2 3
3
9. (1 3 ) 3
34
tt
t t dt t t dt C
53
2 2 4 2
44
10. (2 1) 4 4 1
53
tt
t dt t t dt t C
10
7
3
2
3
3
3
11.
10
y
y y dy y dy C
12. dC
13. 7sin( ) 7cos( )x dx x C
14. 5cos( ) 5sin( )dC
15. 9sin(3 ) 3cos(3 )x dx x C
16. 12cos(4 ) 3sin 4dC
7sin(5 )
17. 7cos(5 )
5
x
x dx C
18. 4sin 12cos
33
xx
dx C
7
7
4
19. 4
7
x
x
e
e dx C
44
20. 9 36
xx
e dx e C
5sin( )
21. 5cos
x
x dx C
6
6
13
22. 13
6
t
t
e
e dt C
II. Evaluate the following definite integrals.
4
3
4
22
1
1
5 188 8
1. (5 8 5) 4 5 60
3 3 3
x
x x dx x x
3
2
9
5
2
9
2
1
1
2 1026 22 1001
2. ( 2 3) 3 200.2
5 5 5 5
x
x x dx x x
9
31
9
22
4
4
1 2 2 20 40
3. ( ) 20 13.333
3 3 3 3
3
x dx x x
x
4
4
32
1
1
5 5 5 5 75
4. 2.344
32 2 32
2
dx
xx
2
34
2
2
1
1
3 44 5 57
5. (1 3 ) 14.25
3 4 3 12 4
tt
t t dt
1
53
1
22
2
2
4 4 7 254 87
6. (2 1) 17.4
5 3 15 15 5
tt
t dt t