The document presents a series of integration problems, including both indefinite and definite integrals, requiring the evaluation of various polynomial and trigonometric expressions. It includes steps and solutions for each integral, as well as specific values calculated for definite integrals. The problems appear structured for practice in basic integration techniques and their applications.

1. Basic Integration Problems
I. Find the following integrals.
2
1. (5 8 5)
x x dx 3 2
2. ( 6 9 4 3)
x x x dx
3
2
3. ( 2 3)
x x dx 2 3
8 5 6
4. dx
x x x
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
x x
2
9. (1 3 )
t t dt 2 2
10. (2 1)
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 18. 4sin
3
x
dx
7
19. 4 x
e dx 4
20. 9
x
e dx
21. 5cos x dx 6
22. 13 t
e dt

2. 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
2 2
2
6. (2 1)
t dt

3. Solutions
I. Find the following integrals.
3
2 2
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
2 3
2 3
1 2
2
8 5 6 8
4. 5 6
5 6 5 3
8 ( ) 8 ( )
1 2
dx x x dx
x x
x x
x x
Ln x Ln x C
x x
1 1
2 2
3 1
3 1
2 2
2 2
1 1
5. ( )
3
3
1 2 2
3 1
3 3 3
2 2
x dx x x dx
x
x x
x x C
8
7
5
3 3
4
3
4
48 27
6. (12 9 )
7 8
x x
x x dx c
2
2
2
4 4
7. 1 4
x
dx x dx x C
x
x
3
2
1 2
8. dx x dx C
x x x
3 4
2 2 3 3
9. (1 3 ) 3
3 4
t t
t t dt t t dt C
5 3
2 2 4 2 4 4
10. (2 1) 4 4 1
5 3
t t
t dt t t dt t C

4. 10
7 3
2 3
3
3
11.
10
y
y y dy y dy C 12. d C
13. 7sin( ) 7cos( )
x dx x C 14. 5cos( ) 5sin( )
d C
15. 9sin(3 ) 3cos(3 )
x dx x C 16. 12cos(4 ) 3sin 4
d C
7sin(5 )
17. 7cos(5 )
5
x
x dx C 18. 4sin 12cos
3 3
x x
dx C
7
7 4
19. 4
7
x
x e
e dx C 4 4
20. 9 36
x x
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
2 2
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
3 1
9
2 2
4
4
1 2 2 20 40
3. ( ) 20 13.333
3 3 3 3
3
x dx x x
x
4
4
3 2
1 1
5 5 5 5 75
4. 2.344
32 2 32
2
dx
x x
2
3 4
2
2
1
1
3 44 5 57
5. (1 3 ) 14.25
3 4 3 12 4
t t
t t dt
1
5 3
1
2 2
2
2
4 4 7 254 87
6. (2 1) 17.4
5 3 15 15 5
t t
t dt t