NCERT Solutions for Class 12 Maths Exercise Miscellaneous hapter 7 Integrals – FREE PDF Download
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NCERT Solutions for Class 12 Maths Chapter 7 Integrals (Ex Misc) Exercise Miscellaneous
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 I1 = 
Putting 
Limits of integration when 
and when 
I1 = 
= 
= 
Putting value of I1 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:
is:(A) 1
(B) 2
(C) 
(D) 
Ans. Let I = 
= 
= 
I =
= 
I = 
= 
I = 
= 0
Therefore, option (B) is correct.