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## NCERT Solutions Class 12 Mathematics Part III . 627 Pages (1343-1970). Free Flip-Book by Study Innovations

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# NCERT Solutions Class 12 Mathematics Part III . 627 Pages (1343-1970). Free Flip-Book by Study Innovations

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Class XII Chapter 7 – Integrals Maths

Also,

Hence, the correct Answer is A.

Page 11 of 216

Class XII Chapter 7 – Integrals Maths
Exercise 7.2
Question 1:

Let = t
∴2x dx = dt

Question 2:
Let log |x| = t

Page 12 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:
Let 1 + log x = t

Question 4:
sin x ⋅ sin (cos x)

Page 13 of 216

Class XII Chapter 7 – Integrals Maths
sin x ⋅ sin (cos x)

Let cos x = t
∴ −sin x dx = dt

Question 5:

Let

Page 14 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:

Let ax + b = t

Question 7:

Let

Page 15 of 216

Class XII Chapter 7 – Integrals Maths
∴ dx = dt

Question 8:
Let 1 + 2x2 = t
∴ 4xdx = dt

Page 16 of 216

Class XII Chapter 7 – Integrals Maths

Question 9:

Let
∴ (2x + 1)dx = dt

Page 17 of 216

Class XII Chapter 7 – Integrals Maths
Question 10:

Let

Question 11:
Let
∴ dx = dt

Page 18 of 216

Class XII Chapter 7 – Integrals Maths

Question 12:
Let

Page 19 of 216

Class XII Chapter 7 – Integrals Maths

Question 13:
Let
∴ 9x2 dx = dt

Page 20 of 216

Class XII Chapter 7 – Integrals Maths

Question 14:

Let log x = t

Question 15:

Page 21 of 216

Class XII Chapter 7 – Integrals Maths
Let
∴ −8x dx = dt

Question 16:

Let

∴ 2dx = dt

Page 22 of 216

Class XII Chapter 7 – Integrals Maths
Question 17:

Let
∴ 2xdx = dt

Question 18:

Let

Page 23 of 216

Class XII Chapter 7 – Integrals Maths

Question 19:
Dividing numerator and denominator by ex, we obtain

Let

Page 24 of 216

Class XII Chapter 7 – Integrals Maths
Question 20:

Let

Question 21:
Let 2x − 3 = t
∴ 2dx = dt

Page 25 of 216

Class XII Chapter 7 – Integrals Maths

Question 22:

Let 7 − 4x = t

∴ −4dx = dt

Question 23:

Let

Page 26 of 216

Class XII Chapter 7 – Integrals Maths

Question 24:

Let

Page 27 of 216

Class XII Chapter 7 – Integrals Maths
Question 25:

Let

Question 26:

Let

Page 28 of 216

Class XII Chapter 7 – Integrals Maths

Question 27:

Let sin 2x = t

Question 28:

Let

Page 29 of 216

Class XII Chapter 7 – Integrals Maths
∴ cos x dx = dt

Question 29:
cot x log sin x
Let log sin x = t

Question 30:

Let 1 + cos x = t

Page 30 of 216

Class XII Chapter 7 – Integrals Maths
∴ −sin x dx = dt

Question 31:

Let 1 + cos x = t
∴ −sin x dx = dt

Question 32:

Page 31 of 216

Class XII Chapter 7 – Integrals Maths

Let sin x + cos x = t ⇒ (cos x − sin x) dx = dt

Question 33:

Page 32 of 216

Class XII Chapter 7 – Integrals Maths

Put cos x − sin x = t ⇒ (−sin x − cos x) dx = dt

Question 34:

Page 33 of 216

Class XII Chapter 7 – Integrals Maths

Question 35:
Let 1 + log x = t

Page 34 of 216

Class XII Chapter 7 – Integrals Maths
Question 36:

Let

Question 37:

Let x4 = t
∴ 4x3 dx = dt

Page 35 of 216

Class XII Chapter 7 – Integrals Maths

Let

From (1), we obtain

Question 38:

equals

Let

Page 36 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is D.
Question 39:

equals

A.
B.
C.
D.

Hence, the correct Answer is B.
Page 37 of 216

Class XII Chapter 7 – Integrals Maths
Exercise 7.3
Question 1:

Question 2:
It is known that,

Page 38 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:
cos 2x cos 4x cos 6x

It is known that,

Question 4:
sin3 (2x + 1)
Let

Page 39 of 216

Class XII Chapter 7 – Integrals Maths

Question 5:
sin3 x cos3 x

Question 6:
sin x sin 2x sin 3x

Page 40 of 216

Class XII Chapter 7 – Integrals Maths
It is known that,

Question 7:
sin 4x sin 8x
It is known that,

Page 41 of 216

Class XII Chapter 7 – Integrals Maths

Question 8:

Question 9:

Page 42 of 216

Class XII Chapter 7 – Integrals Maths

Question 10:
sin4 x

Page 43 of 216

Class XII Chapter 7 – Integrals Maths

Question 11:
cos4 2x

Page 44 of 216

Class XII Chapter 7 – Integrals Maths

Question 12:

Page 45 of 216

Class XII Chapter 7 – Integrals Maths
Question 13:

Question 14:

Page 46 of 216

Class XII Chapter 7 – Integrals Maths

Question 15:

Page 47 of 216

Class XII Chapter 7 – Integrals Maths

Question 16:
tan4x

Page 48 of 216

Class XII Chapter 7 – Integrals Maths
From equation (1), we obtain

Question 17:

Question 18:

Question 19:

Page 49 of 216

Class XII Chapter 7 – Integrals Maths

Question 20:

Page 50 of 216

Class XII Chapter 7 – Integrals Maths

Question 21:
sin−1 (cos x)

Page 51 of 216

Class XII Chapter 7 – Integrals Maths

It is known that,
Substituting in equation (1), we obtain

Page 52 of 216

Class XII Chapter 7 – Integrals Maths

Question 22:

Page 53 of 216

Class XII Chapter 7 – Integrals Maths

Question 23:

is equal to
A. tan x + cot x + C
B. tan x + cosec x + C
C. − tan x + cot x + C
D. tan x + sec x + C

Hence, the correct Answer is A.

Question 24:

equals
A. − cot (exx) + C
B. tan (xex) + C
C. tan (ex) + C
D. cot (ex) + C

Let exx = t

Page 54 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is B.

Page 55 of 216

Class XII Chapter 7 – Integrals Maths
Exercise 7.4
Question 1:

Let x3 = t
∴ 3x2 dx = dt

Question 2:

Let 2x = t
∴ 2dx = dt

Page 56 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:

Let 2 − x = t
⇒ −dx = dt

Question 4:

Let 5x = t

Page 57 of 216

Class XII Chapter 7 – Integrals Maths
∴ 5dx = dt

Question 5:

Question 6:

Let x3 = t

Page 58 of 216

Class XII Chapter 7 – Integrals Maths
∴ 3x2 dx = dt

Question 7:

From (1), we obtain
Page 59 of 216

Class XII Chapter 7 – Integrals Maths

Question 8:

Let x3 = t
⇒ 3x2 dx = dt

Question 9: