Basic Integration Problems Mathematics Calculus.pdf

This is a document with Basic Integration questions

The questions focus on Power rule

1. 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
xxx
1
5. ( )
3
x dx
x
53
34
6. (12 9 )x x dx
2
2
4
7.
x
dx
x
1
8. dx
xx

2
9. (1 3 )t t dt
22
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
22
2
6. (2 1)t dt

3. 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
22
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
xxxx
xx
Ln x Ln x C
xx

11
22
31
31
22
22
11
5. ( )
33
1 2 2
313 3 3
22
x dx x x dx
x
xx
x x C

87
53
34
34
48 27
6. (12 9 )
78
xx
x x dx c

2
2
2
44
7. 1 4
x
dx x dx x C
xx

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

4. 10
7
3
2 33
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
319
22
4
4
1 2 2 20 40
3. ( ) 20 13.333
3 3 3 33
x dx x x
x

44
32
1 1
5 5 5 5 75
4. 2.344
32 2 322
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