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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

Using equations (2) and (3) in (1), we obtain

Question 16:

Let
Let

It is known that,

Question 17:

Page 111 of 216

Class XII Chapter 7 – Integrals Maths

Let

Let ⇒
It is known that,
Question 18:

Page 112 of 216

Class XII Chapter 7 – Integrals Maths

Let ⇒
It is known that,
From equation (1), we obtain

Question 19:

Page 113 of 216

Class XII Chapter 7 – Integrals Maths

Also, let ⇒

It is known that,

Question 20:

Let ⇒
It is known that,

Question 21:

Page 114 of 216

Class XII Chapter 7 – Integrals Maths

Let
Integrating by parts, we obtain

Again integrating by parts, we obtain

Question 22:

Let

Page 115 of 216

Class XII Chapter 7 – Integrals Maths

= 2θ

Integrating by parts, we obtain

Question 23:
equals

Let
Also, let

Page 116 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is A.

Question 24:

equals

Let ⇒
Also, let
It is known that,

Hence, the correct Answer is B.

Page 117 of 216

Class XII Chapter 7 – Integrals Maths
Exercise 7.7
Question 1:

Question 2:

Page 118 of 216

Class XII Chapter 7 – Integrals Maths
Question 3:

Question 4:

Question 5:

Page 119 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:

Question 7:

Page 120 of 216

Class XII Chapter 7 – Integrals Maths

Question 8:

Page 121 of 216

Class XII Chapter 7 – Integrals Maths

Question 9:

Question 10:
is equal to

A.
B.
C.

Page 122 of 216

Class XII Chapter 7 – Integrals Maths

D.

Hence, the correct Answer is A.
Question 11:

is equal to
A.
B.
C.
D.

Hence, the correct Answer is D.
Page 123 of 216

Class XII Chapter 7 – Integrals Maths
Exercise 7.8
Question 1:

It is known that,

Page 124 of 216

Class XII Chapter 7 – Integrals Maths
Question 2:
It is known that,

Question 3:

Page 125 of 216

Class XII Chapter 7 – Integrals Maths

It is known that,

Page 126 of 216

Class XII Chapter 7 – Integrals Maths
Question 4:

It is known that,

Page 127 of 216

Class XII Chapter 7 – Integrals Maths

From equations (2) and (3), we obtain
Page 128 of 216

Class XII Chapter 7 – Integrals Maths

Question 5:
It is known that,

Page 129 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:
It is known that,

Page 130 of 216

Class XII Chapter 7 – Integrals Maths

Page 131 of 216

Class XII Chapter 7 – Integrals Maths
Question 1: Exercise 7.9

By second fundamental theorem of calculus, we obtain

Question 2:

By second fundamental theorem of calculus, we obtain

Question 3:

Page 132 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 4:

By second fundamental theorem of calculus, we obtain
Page 133 of 216

Class XII Chapter 7 – Integrals Maths

Question 5:

By second fundamental theorem of calculus, we obtain

Question 6:

Page 134 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 7:

By second fundamental theorem of calculus, we obtain

Question 8:

Page 135 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 9:

By second fundamental theorem of calculus, we obtain

Question 10:

Page 136 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 11:

By second fundamental theorem of calculus, we obtain

Question 12:

Page 137 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 13:
By second fundamental theorem of calculus, we obtain

Page 138 of 216

Class XII Chapter 7 – Integrals Maths
Question 14:

By second fundamental theorem of calculus, we obtain

Question 15:

Page 139 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 16:
Let

Page 140 of 216

Class XII Chapter 7 – Integrals Maths

Equating the coefficients of x and constant term, we obtain
A = 10 and B = −25

Substituting the value of I1 in (1), we obtain
Page 141 of 216

Class XII Chapter 7 – Integrals Maths

Question 17:

By second fundamental theorem of calculus, we obtain

Question 18:

Page 142 of 216

Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain
Question 19:

By second fundamental theorem of calculus, we obtain

Page 143 of 216

Class XII Chapter 7 – Integrals Maths

Question 20:

By second fundamental theorem of calculus, we obtain
Page 144 of 216

Class XII Chapter 7 – Integrals Maths

Question 21:
equals

A.
B.
C.
D.
By second fundamental theorem of calculus, we obtain

Hence, the correct Answer is D.
Page 145 of 216

Class XII Chapter 7 – Integrals Maths
Question 22:

equals
A.
B.
C.
D.

By second fundamental theorem of calculus, we obtain
Page 146 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is C.

Page 147 of 216

Class XII Chapter 7 – Integrals Maths
Question 1: Exercise 7.10

When x = 0, t = 1 and when x = 1, t = 2

Question 2:
Also, let

Page 148 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:
Also, let x = tanθ ⇒ dx = sec2θ dθ
When x = 0, θ = 0 and when x = 1,

Page 149 of 216

Class XII Chapter 7 – Integrals Maths

Takingθas first function and sec2θ as second function and integrating by parts, we obtain

Question 4:

Let x + 2 = t2 ⇒ dx = 2tdt

When x = 0, and when x = 2, t = 2

Page 150 of 216

Class XII Chapter 7 – Integrals Maths

Question 5:
Let cos x = t ⇒ −sinx dx = dt
When x = 0, t = 1 and when

Page 151 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:

Let ⇒ dx = dt
Page 152 of 216

Class XII Chapter 7 – Integrals Maths

Page 153 of 216

Class XII Chapter 7 – Integrals Maths
Question 7:

Let x + 1 = t ⇒ dx = dt
When x = −1, t = 0 and when x = 1, t = 2

Question 8:
Let 2x = t ⇒ 2dx = dt
When x = 1, t = 2 and when x = 2, t = 4

Page 154 of 216

Class XII Chapter 7 – Integrals Maths

Question 9: is

The value of the integral
A. 6
B. 0
C. 3
D. 4

Page 155 of 216

Class XII Chapter 7 – Integrals Maths

Let cotθ = t ⇒ −cosec2θ dθ= dt
Page 156 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is A.

Question 10:
If
A. cos x + x sin x
B. x sin x
C. x cos x
D. sin x + x cos x

Integrating by parts, we obtain

Page 157 of 216

Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is B.

Page 158 of 216

Class XII Chapter 7 – Integrals Maths
Question 1: Exercise 7.11

Adding (1) and (2), we obtain

Question 2:

Page 159 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 3: