Exercise 7.4
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NCERT Solutions class 12 Maths Integrals
Integrate the following functions in Exercises 1 to 9.
1. 
Ans. Let I = 
= 
……….(i)
Putting 
From eq. (i),
I = 
= 
= 
2. 
Ans. 
= 
= 
= 
3. 
Ans. 
= 
= 
= 
4. 
Ans. 
= 
= 
= 
5. 
Ans. Let I = 
= 
……….(i)
Putting 
From eq. (i),
I = 
= 
= 
= 
= 
6. 
Ans. Let I = 
= 
= 
……….(i)
Putting
From eq. (i),
I = 
= 
= 
= 
7. 
Ans. Let I = 
= 
= 
= 
Let I1 = 
Putting 
I1 = 
= 
= 
= 
I = 
= 
I = 
where 
8. 
Ans. Let I = 
= 
……….(i)
Putting
From eq. (i),
I = 
= 
= 
= 
9. 
Ans. Let I = 
……….(i)
Putting 
From eq. (i),
I = 
= 
= 
= 
Integrate the following functions in Exercises 10 to 18.
10. 
Ans. 
= 
= 
= 
= 
11. 
Ans. 
= 
= 
= 
[For making completing the squares]
= 
= 
= 
= 
= 
= 
12. 
Ans. 
= 
= 
= 
= 
= 
= 
= 
13. 
Ans. 
= 
= 
= 
= 
= 
= 
14. 
Ans. 
= 
= 
= 
= 
= 
= 
= 
15. 
Ans. 
= 
= 
= 
= 
= 
= 
= 
= 
= 
16. 
Ans. Let I = 
……….(i)
Putting 
From eq. (i),
I =
=
= 
= 
= 
17. 
Ans. Let I = 
= 
= 
= 
…….(i)
Let I1 = 
= 
Putting
I1 = 
= 
= 
= 
= 
Putting this value in eq. (i),
NCERT Solutions class 12 Maths Exercise 7.4
18. 
Ans. Let I = 
……….(i)
Let Linear = A 
(Quadratic) + B
……(ii)
Comparing coefficients of 
6A = 5
A = 
Comparing constants,
2A + B = 
On solving, we get A = , B = 
Putting the values of A and B in eq. (ii),
Putting this value of 
in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 = 
= 
= 
……….(iv)
Again I2 =
= 
= 
= 
= 
= 
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
NCERT Solutions class 12 Maths Exercise 7.4
Integrate the following functions in Exercises 19 to 23.
19. 
Ans. Let I = 
= 
……….(i)
Let Linear = A (Quadratic) + B
……(ii)
Comparing coefficients of 2A = 6 A = 3
Comparing constants, –9A + B = 7
On solving, we get A = 3, B = 34
Putting the values of A and B in eq. (ii),
Putting this value of 
in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 = 
=
= 
=
= 
……….(iv)
Again I2 =
= 
= 
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
20. 
Ans. Let I = 
……….(i)
Let Linear = A (Quadratic) + B
……(ii)
Comparing coefficients of –2A = 1 A = 
Comparing constants, 4A + B = 2
On solving, we get A = 
B = 4
Putting the values of A and B in eq. (ii),
Putting this value of 
in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 = = 
= = 
= 
…….(iv)
Again I2 =
= 
= 
= 
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
21. 
Ans. Let I = 
Let Linear = A 
(Quadratic) + B
……(ii)
Comparing coefficients of 2A = 1 A = 
Comparing constants, 2A + B = 2
On solving, we get A = 
B = 1
Putting the values of A and B in eq. (ii),
Putting this value of in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 = =
= =
= 
……….(iv)
Again I2 =
= 
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
22. 
Ans. Let I = 
……….(i)
Let Linear = A (Quadratic) + B
……(ii)
Comparing coefficients of 2A = 1
A =
Comparing constants, –2A + B = 3
On solving, we get A = B = 4
Putting the values of A and B in eq. (ii), 
Putting this value of 
in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 = 
= 
= 
……….(iv)
Again I2 =
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
NCERT Solutions class 12 Maths Exercise 7.4
23. 
Ans. Let I = 
……….(i)
Let Linear = A (Quadratic) + B
……(ii)
Comparing coefficients of 2A = 5 A = 
Comparing constants, 4A + B = 2
On solving, we get A = B = 
Putting the values of A and B in eq. (ii),
Putting this value of 
in eq. (i),
I = 
I = 
I = 
……….(iii)
Now I1 = 
Putting 
I1 =
=
= =
= 
……….(iv)
Again I2 =
= 
= 
= 
= 
……….(v)
Putting values of I1 and I2 in eq. (iii),
I = 
NCERT Solutions class 12 Maths Exercise 7.4
Choose the correct answer in Exercise 24 and 25.
24. 
equals
(A) 
(B) 
(C) 
(D) 
Ans.
= 
= 
=
Therefore, option (B) is correct.
25. 
equals
(A) 
(B) 
(C) 
(D) 
Ans. Let I =
= 
= 
= 
= 
= 
= 
= 
=
Therefore, option (B) is correct.