NCERT Solutions class 12 Maths Miscellaneous Part 2 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Integrals as PDF.**

## 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 =

=

=

**Evaluate the definite integrals in Exercise 25 to 33.**

**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)

Adding eq. (i) and (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 I_{1} =

Putting

Limits of integration when and when

I_{1} =

=

=

Putting value of I_{1 }in 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.

### NCERT Solutions class 12 Maths Miscellaneous Part 2

**43. Choose the correct answer:**

**If then is equal to:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: ……….(i)

Let I = ……….(ii)

I =

= ……….(iii)

Adding eq. (ii) and (iii),

2I =

=

=

I =

Therefore, option (D) is correct.

### NCERT Solutions class 12 Maths Miscellaneous Part 2

**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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