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## NCERT Solutions class 12 Maths Integrals

Evaluate the integrals in Exercises 23 and 24.

23.

Ans. Let I =  ……….(i)

Putting

And

=

=

=  =

I =

=

=

[Applying Product Rule]

=

=

=

=

=

### 24.

Ans. Let I =

=

=

=

=

Putting

I =

=

I =

[Applying Product Rule]

I =

=

I =

=

=

25.

Ans. Let I =

=

=

=

=

=

=

=  =

### 26.

Ans. Let I =

I =

[Dividing each term by ]

I =

=  ……….(i)

Putting

Limits of integration  when  and when

I =

=

=

=

### 27.

Ans. Let I =

=

[Dividing each term by ] ….(i)

Putting

=

Limits of integration  when  and when

I =

=  ……….(ii)

I =

=

I =

=

I =

=

I =

I =

=

### 28.

Ans. Let I = ……….(i)

Putting

Again

Limits of integration  when  (say)

where ……….(ii)

when

I =  =

=  = [From eq. (ii)

29.

Ans. Let I =

=

=

I =

=

=

I =

=

I =

30.

Ans. Let I =

Putting

Again

Limits of integration  when  and

when

I =

=

=

=

I =

=

I =

=

=

31.

Ans. Let I =

=

Putting

Limits of integration  when  and when

I =

=

I =

[Applying Product Rule]

I =

=

I =

=

I =

=  =  =

32.

Ans. Let I =

=

=  ……….(i)

I =

= ……….(ii)

2I =

=

=

=

=

=

=

=

=

=

=

=

=

=  =

I =

33.

Ans. Let I = ……….(i)

If we get

I =

=

I =

I =

=

I =

I =

=

=

Prove the following (Exercise 34 to 40).

34.

Ans. Let I =

=  ……….(i)

Let  ……….(ii)

Comparing coefficients of  A + C + 0 ……….(iii)

Comparing coefficients of  A + B = 0 ……….(iv)

Comparing constants B = 1

On solving eq. (iii), (iv) and (v), we get  A =  B = 1, C = 1

Putting these values of A, B and C in eq. (ii),

I =

=

I =

=

I =

=

I =

=

=

=

35.

Ans. Let I =

=

[Applying Product rule]

I =

=

=  = 1

36.

Ans. Let I =

Here

=

is an odd function of

I =  = 0

37.

Ans. Let I =

=

=

I =

=

I =

=

=

38.

Ans. Let I =

=

=

I =

= ……….(i)

Let I1 =

Putting

Limits of integration  when  and when

I1 =

=

=

Putting value of Iin eq. (i),

I =

=

=

I =

=

=

=

### 39.

Ans. Let I =

Putting

Limits of integration  when  and when , i.e.,

I =

=

=

[Integrating by parts]

I =

=

=

=

### 40. Evaluate  as a limit of sum.

Ans. Given:

Comparing  we have,

Putting these values in

I =

=

I =

=

=

I =

=

### 41. Choose the correct answer:

is equal to:

(A)

(B)

(C)

(D)

Ans. Let I =

=

=

= ……….(i)

Putting

From eq. (i), I =

=

=

Therefore, option (A) is correct.

### 42. Choose the correct answer:

is equal to:

(A)

(B)

(C)

(D)

Ans. Let I =

=

=

I =

=

Therefore, option (B) is correct.

### 43. Choose the correct answer:

If  then  is equal to:

(A)

(B)

(C)

(D)

Ans. Given:  ……….(i)

Let I =  ……….(ii)

I =

=  ……….(iii)

2I =

=

=

I =

Therefore, option (D) is correct.

### 44. The value of  is:

(A) 1

(B) 2

(C)

(D)

Ans. Let I =

=

=

I =

=

I =

=

I =  = 0

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous Part 2

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