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# NCERT Solutions class 12 Maths Miscellaneous Part 2

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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

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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