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NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

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NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

NCERT Solutions Class 12 Mathematics Part II . 674 Pages (668-1342). Free Flip-Book by Study Innovations

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Class XII Chapter 7 – Integrals Maths
Exercise 7.6

Question 1:
x sin x
Answer

Let I =
Taking x as first function and sin x as second function and integrating by parts, we
obtain

Question 2:

Answer

Let I =
Taking x as first function and sin 3x as second function and integrating by parts, we
obtain

Question 3:

Page 101 of 216


Class XII Chapter 7 – Integrals Maths
Answer

Let
Taking x2 as first function and ex as second function and integrating by parts, we obtain

Again integrating by parts, we obtain

Question 4:
x logx
Answer

Let
Taking log x as first function and x as second function and integrating by parts, we
obtain

Page 102 of 216


Class XII Chapter 7 – Integrals Maths

Question 5:
x log 2x
Answer

Let
Taking log 2x as first function and x as second function and integrating by parts, we
obtain

Question 6:
x2 log x
Answer

Let
Taking log x as first function and x2 as second function and integrating by parts, we
obtain

Page 103 of 216


Class XII Chapter 7 – Integrals Maths
Question 7:

Answer as first function and x as second function and integrating by parts, we

Let
Taking
obtain

Question 8:

Answer
Let

Page 104 of 216


Class XII Chapter 7 – Integrals Maths

Taking as first function and x as second function and integrating by parts, we
obtain

Question 9:
Answer
Let
Taking cos−1 x as first function and x as second function and integrating by parts, we
obtain

Page 105 of 216


Class XII Chapter 7 – Integrals Maths

Page 106 of 216


Class XII Chapter 7 – Integrals Maths
Question 10:

Answer as first function and 1 as second function and integrating by parts, we

Let

Taking
obtain

Question 11:
Answer
Let

Page 107 of 216


Class XII Chapter 7 – Integrals Maths

Taking as first function and as second function and integrating by parts,
we obtain

Question 12:

Answer

Let
Taking x as first function and sec2x as second function and integrating by parts, we
obtain

Question 13:
Answer

Page 108 of 216


Class XII Chapter 7 – Integrals Maths

Let as first function and 1 as second function and integrating by parts, we

Taking
obtain

Question 14:
Answer

Taking as first function and 1 as second function and integrating by parts, we
obtain

Again integrating by parts, we obtain
Page 109 of 216


Class XII Chapter 7 – Integrals Maths

Question 15:

Answer

Let
Let I = I1 + I2 … (1)

Where, and

Taking log x as first function and x2 as second function and integrating by parts, we
obtain

Taking log x as first function and 1 as second function and integrating by parts, we
obtain

Page 110 of 216


Class XII Chapter 7 – Integrals Maths

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

Question 16:

Answer
Let
Let


It is known that,

Question 17:

Page 111 of 216


Class XII Chapter 7 – Integrals Maths
Answer

Let

Let ⇒
It is known that,
Question 18:
Answer

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
Answer

Also, let ⇒

It is known that,

Question 20:
Answer

Let ⇒
It is known that,

Question 21:
Answer

Page 114 of 216


Class XII Chapter 7 – Integrals Maths

Let
Integrating by parts, we obtain

Again integrating by parts, we obtain

Question 22:

Answer ⇒
Let

Page 115 of 216


Class XII Chapter 7 – Integrals Maths

= 2θ


Integrating by parts, we obtain

Question 23:
equals

Answer ⇒

Let
Also, let

Page 116 of 216


Class XII Chapter 7 – Integrals Maths

Hence, the correct Answer is A.

Question 24:

equals

Answer

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:
Answer

Question 2:
Answer

Page 118 of 216


Class XII Chapter 7 – Integrals Maths
Question 3:

Answer

Question 4:
Answer

Question 5:

Page 119 of 216


Class XII Chapter 7 – Integrals Maths
Answer

Question 6:
Answer

Question 7:
Answer

Page 120 of 216


Class XII Chapter 7 – Integrals Maths

Question 8:
Answer

Page 121 of 216


Class XII Chapter 7 – Integrals Maths

Question 9:
Answer

Question 10:
is equal to

A.
B.
C.

Page 122 of 216


Class XII Chapter 7 – Integrals Maths

D.
Answer

Hence, the correct Answer is A.
Question 11:

is equal to
A.
B.
C.
D.
Answer

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


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

Answer
It is known that,

Page 124 of 216


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

Question 3:

Page 125 of 216


Class XII Chapter 7 – Integrals Maths

Answer
It is known that,

Page 126 of 216


Class XII Chapter 7 – Integrals Maths
Question 4:
Answer

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:
Answer
It is known that,

Page 129 of 216


Class XII Chapter 7 – Integrals Maths

Question 6:
Answer
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
Answer

By second fundamental theorem of calculus, we obtain

Question 2:
Answer

By second fundamental theorem of calculus, we obtain

Question 3:

Page 132 of 216


Class XII Chapter 7 – Integrals Maths
Answer

By second fundamental theorem of calculus, we obtain

Question 4:
Answer

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


Class XII Chapter 7 – Integrals Maths

Question 5:
Answer

By second fundamental theorem of calculus, we obtain

Question 6:
Answer

Page 134 of 216


Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 7:
Answer

By second fundamental theorem of calculus, we obtain

Question 8:
Answer

Page 135 of 216


Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 9:
Answer

By second fundamental theorem of calculus, we obtain

Question 10:

Page 136 of 216


Class XII Chapter 7 – Integrals Maths
Answer

By second fundamental theorem of calculus, we obtain

Question 11:
Answer

By second fundamental theorem of calculus, we obtain

Question 12:

Page 137 of 216


Class XII Chapter 7 – Integrals Maths
Answer

By second fundamental theorem of calculus, we obtain

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

Page 138 of 216


Class XII Chapter 7 – Integrals Maths
Question 14:
Answer

By second fundamental theorem of calculus, we obtain

Question 15:
Answer

Page 139 of 216


Class XII Chapter 7 – Integrals Maths

By second fundamental theorem of calculus, we obtain

Question 16:
Answer
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:
Answer

By second fundamental theorem of calculus, we obtain

Question 18:
Answer

Page 142 of 216


Class XII Chapter 7 – Integrals Maths

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

By second fundamental theorem of calculus, we obtain

Page 143 of 216


Class XII Chapter 7 – Integrals Maths

Question 20:
Answer

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.
Answer
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.
Answer

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
Answer

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

Question 2:
Answer
Also, let

Page 148 of 216


Class XII Chapter 7 – Integrals Maths

Question 3:
Answer
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:

Answer

Let x + 2 = t2 ⇒ dx = 2tdt

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

Page 150 of 216


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