7. INTEGRALS

Class 12 Mathematics Chapter 7. INTEGRALS

Chapter 7: Integrals

Exercise 7.1

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

1. 2. 3. 4. 5.

Solution : 1.

We have,

Solution : 2.

We have,

Solution : 3.

We have,

Solution : 4.

We have,

Solution : 5.

We have,

Find the following integrals in Exercises 6 to 20 .

Solution: We have,

10.

Solution: We have ,

11.

Solution: We have ,

12.

Solution: We have,

13.

Solution: We have ,

14.

Solution: We have,

15.

Solution: We have,

16.

Solution : We have ,

17.

Solution: We have,

18.

Solution: We have,

19.

Solution: We have,

20.

Solution: We have,

21. The anti-derivative of equals

(A) (B)

Solution: We have,

The correct answer (C) .

22. If such that. The is :

(A) (B)

Solution: We have,

From (i) , we get

The correct answer : (A)

Exercise 7.2

Integrate the functions in Exercises 1 to 37 :

Solution: We have,

Solution : We have,

Let

Now ,

Solution: We have,

Let

Now ,

Solution : We have,

Let

Now,

Solution: We have,

Let

Now,

Solution: We have,

Let

Now,

Solution: We have,

Let

Now,

Solution : We have,

Let

Now

Solution: We have,

10.

Solution: We have,

Let

11.

Solution: We have,

12.

Solution: We have,

Let

Now ,

13.

Solution: We have,

Let

Now ,

14.

Solution: We have,

Let

Now ,

15.

Solution: We have,

Let

Now

16.

Solution : We have,

Let

Now

17.

Solution: We have,

Let

18.

Solution: We have,

Let

Now ,

19.

Solution : We have,

20.

Solution: We have,

21.

Solution: We have,

Let

Now

22.

Solution: We have,

Let

23.

Solution: We have,

24.

Solution : We have,

25.

Solution: We have,

26.

Solution: We have,

27.

Solution: We have,

28.

Solution: We have,

29.

Solution : We have,

Let

Now

30.

Solution : We have,

31.

Solution: We have,

32.

Solution: We have,

33.

Solution: We have,

Let

Now ,

34.

Solution : We have,

Let

Now ,

35.

Solution : We have,

Let

Now ,

36.

Solution : We have,

Let

Now,

37.

Solution : We have,

Let

Now ,

38. equals

(A) (B)

Solution : We have,

Let

Now,

The correct Option (D)

40. equals

(A) (B)

Solution : We have,

The correct option (B)

Exercise 7.3

Find the integrals of the functions in Exercises 1 to 22 :

Solution : We have,

Solution: We have ,

Solution : We have ,

Let

Now

Solution : We have,

Let

Now

Solution : We have,

10.

Solution : We have,

11.

Solution : We have,

12.

Solution : We have,

13.

Solution : We have,

14.

Solution : We have

Let

Now

15.

Solution : We have,

Let

Now

16.

Solution : We have,

Let

Now,

17.

Solution : We have ,

18.

Solution : We have,

19.

Solution : We have,

Let

Now ,

20.

Solution : We have,

Let

Now ,

21.

Solution : We have,

22.

Solution : We have,

Choose the correct answer in Exercises 23 and 24 .

23. is equal to

(A) (B)

Solution : We have,

(A)

24. equals

(A) (B)

Solution : We have,

Let

Now

(B)

Exercise 7.4

Solution : We have,

Let

Now

Solution : We have,

Let

Now ,

Solution : We have,

Let

Now

Solution : We have,

Let

Now ,

Solution : We have,

Let

Now

10.

Solution : We have,

11.

Solution : We have,

Let

Now ,

12.

Solution : We have,

13.

Solution : We have,

14.

Solution : We have,

15.

Solution : We have,

1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16.

17. 18. ² 19. 20. 21. 22. 23.

Choose the correct answer in Exercises 24 and 25 .

24. equals

(A) (B)

25. equals

(A) (B)

Exercise 7.5

Integrate the rational functions in Exercises 1 to 21 .

1. 2. 3. 4. 5. 6. 7.

8. 9. 10. 11. 12. 13. 14.

15. 16. [ Hint : multiply numerator and Denominator by and ]

17. [Hint : Put ]

18. 19. 20. 21. [Hint: ]

Choose the correct answer in each of the Exercises 22 and 23 .

22. equals

(A) (B)

23. equals

(A)

(B)

(C)

(D)

Exercise 7.6

Integrate the functions in Exercises 1 to 22 :

1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21.

22.

Choose the correct answer in Exercises 23 and 24 .

23. equals

(A) (B) (C) (D)

24. equals

(A) (B)

Exercise 7.7

Integrate the functions in Exercises 1 to 9 .

1. 2. 3. 4.

5. 6. 7. 8. 9.

Choose the correct answer in Exercises 10 to 11 .

10. is equals

(A)

(B)

(C)

(D)

11. is equals

(A)

(B)

(C)

(D)

Exercise 7.8

Evaluate the definite integrals in Exercises 1 to 20.

Solution : We have,

Solution : we have,

Solution : We have,

Note :

Solution : We have ,

10.

Solution : We have,

11.

Solution : We have,

12.

Solution : We have,

13.

Solution : We have,

14.

Solution: We have,

15.

Solution : We have,

16.

Solution : We have,

Let,

Putting , then

Now,

17.

Solution : We have,

18.

Solution: We have,

19.

Solution : We have,

20.

Solution: We have,

Choose the correct answer in Exercises 21 to 22 .

21. equals

(A) (B) (C) (D)

Solution: We have,

Correct Option : (D)

22. equals

(A) (B) (C) (D)

Solution: We have,

Cottect Option : (C)

Exercise 7.9

Evaluate the integrals in Exercises 1 to 8 using Substitution .

Solution : We have,

Let

Putting 0 ,then

Putting then

Now,

[ Note : ]

Solution : We have,

4. [Put ]

Solution : We have,

Put

Putting , then

and

Now,

Solution : We have,

Let

put and

Now ,

1. 2. 3. 4. [Put x ]

5. 6. 7. 8.

Choose the correct answer in Exercise 9 to 10 .

9. The value of the integral is

(A) 6 (B) 0 (C) 3 (D) 4

10. If f , then is

(A) (B)

Exercise 7.11

By using the properties of definite integrals , evaluate the integrals in Exercises 1 to 19 .

1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11. 12. 13.

14. 15. 16. 17.

18.

19. Show that , if and are defined as and .

Choose the correct answer in Exercises 20 to 21 .

20. The value of is

(A) 0 (B) 2 (C) (D) 1

21. The value of is

(A) 2 (B) (C) 0 (D) – 2

Miscellaneous Exercise on Chapter 7

Integrate the function in Exercises 1 to 24 .

1. 1x-x³ 2. 3. [ Hint : Put ]

4. 5. [Hint : , Put ]

6. 7. 8. hjhjhjhjhjhj