Solution of Chapter 7. Integrals (Mathematics Part-II Book)

Chapter Exercises

Exercise 7.1
Exercise 7.2
Exercise 7.3
Exercise 7.4
Exercise 7.5
Exercise 7.6
Exercise 7.7
Exercise 7.8
Exercise 7.9
Exercise 7.10
Exercise 7.11
Miscellaneous Exercise

Exercise 7.1

1

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x
2

Find an anti-derivative (or integral) of the following functions by the method of inspection.

cos 3x
3

Find an anti-derivative (or integral) of the following functions by the method of inspection.

e2x
4

Find an anti-derivative (or integral) of the following functions by the method of inspection.

(ax + b)2
5

Find an anti-derivative (or integral) of the following functions by the method of inspection.

sin 2x – 4 e3x
6

Find the following integrals.
7

Find the following integrals.
8

Find the following integrals.
9

Find the following integrals.
10

Find the following integrals.
11

Find the following integrals.
12

Find the following integrals.
13

Find the following integrals.
14

Find the following integrals.
15

Find the following integrals.
16

Find the following integrals.
17

Find the following integrals.
18

Find the following integrals.
19

Find the following integrals.
20

Find the following integrals.
21

The anti-derivative of equals
22

If such that f(2) = 0. Then f(x) is

Exercise 7.2

1

Integrate the functions.
2

Integrate the functions.
3

Integrate the functions.
4

Integrate the functions.

sin x sin (cos x)
5

Integrate the functions.

sin (ax + b) cos (ax + b)
6

Integrate the functions.
7

Integrate the functions.
8

Integrate the functions.
9

Integrate the functions.
10

Integrate the functions.
11

Integrate the functions.
12

Integrate the functions.
13

Integrate the functions.
14

Integrate the functions.
15

Integrate the functions.
16

Integrate the functions.
17

Integrate the functions.
18

Integrate the functions.
19

Integrate the functions.
20

Integrate the functions.
21

Integrate the functions.

tan2 (2x – 3)
22

Integrate the functions.

sec2 (7 – 4x)
23

Integrate the functions.
24

Integrate the functions.
25

Integrate the functions.
26

Integrate the functions.
27

Integrate the functions.
28

Integrate the functions.
29

Integrate the functions.

cot x log sin x
30

Integrate the functions.
31

Integrate the functions.
32

Integrate the functions.
33

Integrate the functions.
34

Integrate the functions.
35

Integrate the functions.
36

Integrate the functions.
37

Integrate the functions.
38

39

equals

Exercise 7.3

1

Find the integrals of the functions.
2

Find the integrals of the functions.
1

Find the integrals of the functions.
4

Find the integrals of the functions.
5

Find the integrals of the functions.
6

Find the integrals of the functions.
7

Find the integrals of the functions.
8

Find the integrals of the functions.
9

Find the integrals of the functions.
9

Find the integrals of the functions.
11

Find the integrals of the functions.
12

Find the integrals of the functions.
13

Find the integrals of the functions.
14

Find the integrals of the functions.
1

Find the integrals of the functions.
16

Find the integrals of the functions.
17

Find the integrals of the functions.
18

Find the integrals of the functions.
19

Find the integrals of the functions.
20

Find the integrals of the functions.
21

Find the integrals of the functions.
22

Find the integrals of the functions.
23


A. B.


C. D.

24


A. B.


C. D.

Exercise 7.4

1

Integrate the functions.
2

Integrate the functions.
3

Integrate the functions.
4

Integrate the functions.
5

Integrate the functions.
6

Integrate the functions.
7

Integrate the functions.
8

Integrate the functions.
9

Integrate the functions.
10

Integrate the functions.
11

Integrate the functions.
12

Integrate the functions.
13

Integrate the functions.
14

Integrate the functions.
15

Integrate the functions.
16

Integrate the functions.
17

Integrate the functions.
18

Integrate the functions.
19

Integrate the functions.
20

Integrate the functions.
21

Integrate the functions.
22

Integrate the functions.
23

Integrate the functions.
24

25

Exercise 7.5

1

Integrate the rational functions.
2

Integrate the rational functions.
3

Integrate the rational functions.
4

Integrate the rational functions.
5

Integrate the rational functions.
6

Integrate the rational functions.
7

Integrate the rational functions.
8

Integrate the rational functions.
9

Integrate the rational functions.
10

Integrate the rational functions.
11

Integrate the rational functions.
12

Integrate the rational functions.
13

Integrate the rational functions.
14

Integrate the rational functions.
15

Integrate the rational functions.
16

Integrate the rational functions.

[Hint: multiply numerator and denominator by xn-1 and put xn=t]
17

Integrate the rational functions.
18

Integrate the rational functions.
19

Integrate the rational functions.
20

Integrate the rational functions.
21

Integrate the rational functions.

[Hint: Put ex = t]
22

23

Integrate the rational functions.

Exercise 7.6

1

Integrate the functions.
2

Integrate the functions.
3

Integrate the functions.
4

Integrate the functions.
5

Integrate the functions.
6

Integrate the functions.
7

Integrate the functions.
8

Integrate the functions.
9

Integrate the functions.
10

Integrate the functions.
11

Integrate the functions.
12

Integrate the functions.
13

Integrate the functions.
14

Integrate the functions.
15

Integrate the functions.
16

Integrate the functions.
17

Integrate the functions.
18

Integrate the functions.
19

Integrate the functions.
20

Integrate the functions.
21

Integrate the functions.
22

Integrate the functions.
23

23

Exercise 7.7

1

Integrate :
2

Integrate :
3

Integrate :
4

Integrate :

5

Integrate :

6

Integrate :

7

Integrate :
8

Integrate :

9

Integrate :
10

Choose the correct answer

is equal to
11

Choose the correct answer

is equal to


A.


B.


C.


D.

Exercise 7.8

1

Evaluate the following definite integrals as limit of sums.
2

Evaluate the following definite integrals as limit of sums.
3

Evaluate the following definite integrals as limit of sums.
4

Evaluate the following definite integrals as limit of sums.
5

Evaluate the following definite integrals as limit of sums.
6

Evaluate the following definite integrals as limit of sums.

Exercise 7.9

1

Evaluate

2

Evaluate

3

Evaluate

4

Evaluate

5

Evaluate

6

Evaluate

7

Evaluate

8

Evaluate

9

Evaluate

10

Evaluate

11

Evaluate

12

Evaluate

13

Evaluate

14

Evaluate

15

Evaluate

16

Evaluate

17

Evaluate

18

Evaluate

19

Evaluate

20

Evaluate

21

equals
22

equals

Exercise 7.10

1

Evaluate the integrals using substitution.
2

Evaluate the integrals using substitution.
3

Evaluate the integrals using substitution.
4

Evaluate the integrals using substitution.
5

Evaluate the integrals using substitution.
6

Evaluate the integrals using substitution.
7

Evaluate the integrals using substitution.
8

Evaluate the integrals using substitution.
9

The value of the integral is
10

If then f'(x) is

Exercise 7.11

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Show that if f and g are defined as and

20

The value of

21

The value of

Miscellaneous Exercise

1

Integrate the function:

2

Integrate the function:

3

Integrate the function:

4

Integrate the function:

5

Integrate the function:

6

Integrate the function:

7

Integrate the function:

8

Integrate the function:

9

Integrate the function:

10

Integrate the function:

11

Integrate the function:

12

Integrate the function:

13

Integrate the function:

14

Integrate the function:

15

Integrate the function:

16

Integrate the function:

17

Integrate the function:

18

Integrate the function:

19

Integrate the function:

20

Integrate the function:

21

Integrate the function:

22

Integrate the function:

23

Integrate the function:

24

Integrate:

25

Evaluate the definite integral:
26

Evaluate the definite integral:
27

Evaluate the definite integral:
28

Evaluate the definite integral:

29

Evaluate the definite integral:
30

Evaluate the definite integral:

31

Evaluate the definite integral:
32

Evaluate the definite integral:
33

Evaluate the definite integral:
34

Prove:

35

Prove:

36

Prove:

37

Prove:

38

Prove:

39

Prove:

40

Evaluate as a limit of a sum.
41

Choose the correct answers

is equal to
42

Choose the correct answers

is equal to
43

Choose the correct answers

If f (a + b – x) = f (x), then is equal to
44

Choose the correct answers

The value of