## NCERT Solutions for Class 12 Maths Exercise Miscellaneous hapter 7 Integrals – FREE PDF Download

NCERT Solutions for integrals Class 12 exercises Miscellaneous are accessible in PDF format which is available at CoolGyan’s online learning portal. The solutions to this particular exercise from chapter 7 Integrals have been prepared by our experts who have years of experience in this field. Students can find, well-explained solutions to maths problems at CoolGyan’s online learning portal. All the solutions are provided in a way that will help you in understanding the concept effectively. You can download them in PDF format for free.

# NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex Misc) Exercise Miscellaneous

**Integrate the function in Exercises 1 to 11.**

**1. **

**Ans. **Let I =

Here,

=

=

= ……….(i)

Comparing the coefficients of –A + B – C = 0 ……….(ii)

Comparing the coefficients of B + C = 0 ……….(iii)

Comparing constants A = 1 ……….(iv)

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

Putting these values in eq. (i),

=

=

=

=

=

=

=

**2. **

**Ans. **Let I =

=

=

=

=

=

=

=

=

**3. **

**Ans. **Let I = ……….(i)

Putting

From eq. (i),

I =

=

=

=

=

=

**4. **

**Ans. **Let I =

=

=

=

Putting

I =

=

=

**5. **

**Ans. **Let I =

Putting

I =

=

=

=

=

=

=

=

=

=

**6. **

**Ans. **Let I = ……….(i)

Let ……….(ii)

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

Comparing coefficients of B + C = 5 ……….(iv)

Comparing constants 9A + C = 0 ……….(v)

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

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

From eq. (i),

I =

=

=

=

**7. **

**Ans. **Let I =

=

=

=

=

=

=

=

=

**8. **

**Ans. **Let I =

=

=

=

I =

=

**9. **

**Ans. **Let I = ……….(i)

Putting

From eq. (i),

I =

=

=

**10. **

**Ans. **Let I = ……….(i)

I =

=

I =

I =

I =

=

I =

=

**11. **

**Ans. **Let I = ……….(i)

I =

=

I =

I =

I =

I =

I =

**Integrate the function in Exercises 12 to 22.**

**12. **

**Ans. **Let I =

= ……….(i)

Putting

From eq. (i),

I =

=

=

**13. **

**Ans. **Let I = ……….(i)

Putting

From eq. (i), I =

= ……….(ii)

I =

=

I =

=

I =

=

**14. **

**Ans. **Let I = ……….(i)

I =

=

I =

=

I = +c

**15. **

**Ans. **Let I =

=

= ……….(i)

Putting

From eq. (i),

I =

=

=

**16. **

**Ans. **Let I =

=

=

=

I =

Putting

From eq. (i),

I =

=

I =

=

**17. **

**Ans. **Let I =

= ………..(i)

Putting

From eq. (i),

I =

= if

And I =

=

= if

I = if

And I = if

**18. **

**Ans. **Let I =

=

I =

=

I =

Putting

I =

=

=

I =

=

I =

=

**19. **

**Ans. **We know that

I =

=

I =

= ……….(i)

Putting

=

I =

=

I =

[Applying product rule]

I =

I =

I =

Putting

I =

I =

I =

**20. **

**Ans. **Let I =

Putting

I =

=

=

=

I = ……….(i)

I =

=

I =

I = ……….(ii)

For evaluating , putting

=

=

=

= ……….(iii)

Putting this value in eq. (ii),

I =

I =

=

I =

**21. **

**Ans. **Let I =

=

=

=

=

=

=

**22. **

**Ans. **Let I = ……….(i)

Let ……….(ii)

Comparing coefficients of A + C = 1 ……….(iii)

Comparing coefficients of 3A + B + 2C = 1 ……….(iv)

Comparing constants 2A + 2B + C = 1 ……….(v)

On solving eq. (iii), (iv) and (v), we get A = –2, B = 1, C = 3

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

=

=

=

=