Exercise 7.11
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NCERT Solutions class 12 Maths Integrals
By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 6.
1. 
Ans. Let I = ……….(i)
= 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
= 
2I = 
I = 
2. 
Ans. Let I = 
………..(i)
I = 
I = 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = =
2I =
I =
3. 
Ans. Let I = 
………..(i)
I = 
I = 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = =
2I =
I =
4. 
Ans. Let I = 
………..(i)
I = 
I = 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = =
2I = I =
5. 
Ans. Let I = ……….(i)
Putting 
From eq. (i),
I = 
= 
= 
= 
= 
= 
= 
6. 
Ans. Let I = ……….(i)
Putting 
From eq. (i),
I = 
= 
= 
= 
= 
= 
= 
By using the properties of definite integrals, evaluate the integrals in Exercises 7 to 11.
7. 
Ans. Let I =
= 
I = 
= 
= 
I = 
= 
= 
8. 
Ans. Let I = ……….(i)
I = 
I = 
= 
= 
…..(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = 
= 
= 
I = 
9. 
Ans. Let I =
= 
I = 
= 
= 
= 
I = 
= 
= 
10. 
Ans. Let I =
= 
= 
= 
= 
……….(i)
I = 
I = 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = 
= 
= 
= 
I = 
= 
= 
11. 
Ans. Let I =
= 
…(i)
I = 
I = 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
= 
= 
I =
Using properties of definite integrals, evaluate the following integrals in Exercise 12 to 18.
12. 
Ans. Let I = 
……….(i)
I = 
= 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
= 
= 
2I = 
2I = 
2I = 
I = 
= 
= 
= 
= 
13. 
Ans. Let I =
Here 
= 
= 
is an odd function of 
I = = 0
14. 
Ans. 
= 
Here 
NCERT Solutions class 12 Maths Exercise 7.11
15. 
Ans. Let I = ……….(i)
I = 
= 
= 
….(ii)
Adding eq. (i) and (ii), we have 2I = 0 I = 0
16. 
Ans. Let I = 
……….(i)
I = 
= 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = 
= 
= 
I = 
……….(iii)
I = 
= 
……….(iv)
Adding eq. (i) and (ii),
2I = 
= 
I = 
= 
= 
I = 
= 
= 
I = I1 – 
……….(v)
Where I1 = 
…….(vi)
Putting 
in eq. (vi),
Limits of integration when 
and 
From eq. (vi),
I1 = 
= 
= 
I1 = 
= 
I1 = 
[From eq. (iii)]
Putting this value in eq. (v), I = 
2I = 
2I – I = 
I =
17. 
Ans. Let I = ……….(i)
I = 
= 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
= 
= 
= 
I = 
18. 
Ans. Let I = ……….(i)
Here 
From eq. (i),
I = 
= 
I = 
= 
I = 
= 
= 6 – 1
= 5
NCERT Solutions class 12 Maths Exercise 7.11
19. Show that 
are defined as 
and 
Ans. Here ……..(i) and ………(ii)
Let I = 
……….(iii)
I = 
= 
[From eq. (i)] ……….(iv)
Adding eq. (iii) and (iv)
2I = 
= 
2I = 
[From eq. (ii)]
2I = 
I = 
Hence proved.
NCERT Solutions class 12 Maths Exercise 7.11
Choose the correct answer in Exercises 20 and 21.
20. The value of 
is:
(A) 0
(B) 2
(C) 
(D) 1
Ans. Let I =
= 
I = 
= 
Therefore, option (C) is correct.
NCERT Solutions class 12 Maths Exercise 7.11
21. The value of 
is:
(A) 2
(B) 
(C) 0
(D) 
Ans. Let I = 
……….(i)
I = 
= 
……….(ii)
Adding eq. (i) and (ii),
2I = 
= 
2I = 
= 
= 0
I = 0