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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

Adding (1) and (2), we obtain

Question 4:

Page 161 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 5:
It can be seen that (x + 2) ≤ 0 on [−5, −2] and (x + 2) ≥ 0 on [−2, 5].

Page 162 of 216

Class XII Chapter 7 – Integrals Maths

Question 6:
It can be seen that (x − 5) ≤ 0 on [2, 5] and (x − 5) ≥ 0 on [5, 8].

Question 7:

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Class XII Chapter 7 – Integrals Maths

Question 8:

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Class XII Chapter 7 – Integrals Maths

Question 9:

Page 165 of 216

Class XII Chapter 7 – Integrals Maths

Question 10:

Page 166 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 11:
As sin2 (−x) = (sin (−x))2 = (−sin x)2 = sin2x, therefore, sin2x is an even function.

Page 167 of 216

Class XII Chapter 7 – Integrals Maths

It is known that if f(x) is an even function, then

Question 12:
Adding (1) and (2), we obtain

Page 168 of 216

Class XII Chapter 7 – Integrals Maths

Question 13:

As sin7 (−x) = (sin (−x))7 = (−sin x)7 = −sin7x, therefore, sin2x is an odd function.
It is known that, if f(x) is an odd function, then

Question 14:

It is known that,

Page 169 of 216

Class XII Chapter 7 – Integrals Maths

Question 15:

Adding (1) and (2), we obtain
Question 16:

Page 170 of 216

Class XII Chapter 7 – Integrals Maths
Adding (1) and (2), we obtain

sin (π − x) = sin x
Adding (4) and (5), we obtain

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

Page 171 of 216

Class XII Chapter 7 – Integrals Maths
Question 17:

It is known that,
Adding (1) and (2), we obtain

Question 18:
It can be seen that, (x − 1) ≤ 0 when 0 ≤ x ≤ 1 and (x − 1) ≥ 0 when 1 ≤ x ≤ 4

Page 172 of 216

Class XII Chapter 7 – Integrals Maths

Question 19: if f and g are defined as and
Show that

Adding (1) and (2), we obtain

Question 20:

Page 173 of 216

Class XII Chapter 7 – Integrals Maths
is
The value of
A. 0
B. 2
C. π
D. 1

It is known that if f(x) is an even function, then and
if f(x) is an odd function, then

Hence, the correct Answer is C.
Question 21:

The value of is
A. 2

B.
C. 0
D.

Page 174 of 216

Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Hence, the correct Answer is C.

Page 175 of 216

Class XII Chapter 7 – Integrals Maths
Miscellaneous Solutions
Question 1:

Equating the coefficients of x2, x, and constant term, we obtain
−A + B − C = 0
B+C=0
A=1
On solving these equations, we obtain

From equation (1), we obtain

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Class XII Chapter 7 – Integrals Maths

Question 2:

Page 177 of 216

Class XII Chapter 7 – Integrals Maths

Question 3:

[Hint: Put ]

Page 178 of 216

Class XII Chapter 7 – Integrals Maths

Question 4:

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Class XII Chapter 7 – Integrals Maths

Page 180 of 216

Class XII Chapter 7 – Integrals Maths
Question 5:

On dividing, we obtain

Question 6:

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Class XII Chapter 7 – Integrals Maths

Equating the coefficients of x2, x, and constant term, we obtain
A+B=0
B+C=5
9A + C = 0
On solving these equations, we obtain

From equation (1), we obtain

Question 7:

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Class XII Chapter 7 – Integrals Maths

Let x − a = t ⇒ dx = dt

Question 8:

Question 9:

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Class XII Chapter 7 – Integrals Maths

Let sin x = t ⇒ cos x dx = dt

Question 10:

Question 11:

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Class XII Chapter 7 – Integrals Maths

Question 12:
Let x4 = t ⇒ 4x3 dx = dt

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Class XII Chapter 7 – Integrals Maths

Question 13:
Let ex = t ⇒ ex dx = dt

Question 14:

Page 186 of 216

Class XII Chapter 7 – Integrals Maths

Equating the coefficients of x3, x2, x, and constant term, we obtain
A+C=0
B+D=0
4A + C = 0
4B + D = 1
On solving these equations, we obtain

From equation (1), we obtain

Question 15:

= cos3 x × sin x
Let cos x = t ⇒ −sin x dx = dt

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Class XII Chapter 7 – Integrals Maths

Question 16:

Question 17:

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Class XII Chapter 7 – Integrals Maths

Question 18:

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Class XII Chapter 7 – Integrals Maths

Question 19:

Page 190 of 216

Class XII Chapter 7 – Integrals Maths

From equation (1), we obtain
Page 191 of 216

Class XII Chapter 7 – Integrals Maths

Question 20:

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Class XII Chapter 7 – Integrals Maths

Question 21:

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Class XII Chapter 7 – Integrals Maths

Question 22:

Equating the coefficients of x2, x,and constant term, we obtain
A+C=1
3A + B + 2C = 1
2A + 2B + C = 1
On solving these equations, we obtain
A = −2, B = 1, and C = 3
From equation (1), we obtain

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Class XII Chapter 7 – Integrals Maths

Question 23:

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Class XII Chapter 7 – Integrals Maths
Question 24:

Integrating by parts, we obtain
Page 196 of 216

Class XII Chapter 7 – Integrals Maths

Question 25:

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Class XII Chapter 7 – Integrals Maths

Question 26:

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Class XII Chapter 7 – Integrals Maths

When x = 0, t = 0 and

Question 27:

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Class XII Chapter 7 – Integrals Maths

When and when

Page 200 of 216

Class XII Chapter 7 – Integrals Maths
Question 28:

When and when

As , therefore, is an even function.

It is known that if f(x) is an even function, then

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Class XII Chapter 7 – Integrals Maths

Question 29:

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Class XII Chapter 7 – Integrals Maths
Question 30:

Question 31:

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Class XII Chapter 7 – Integrals Maths

From equation (1), we obtain

Question 32:

Page 204 of 216

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Class XII Chapter 7 – Integrals Maths

Adding (1) and (2), we obtain

Question 33:

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Class XII Chapter 7 – Integrals Maths

Page 206 of 216

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Class XII Chapter 7 – Integrals Maths

From equations (1), (2), (3), and (4), we obtain

Question 34:

Equating the coefficients of x2, x, and constant term, we obtain
A+C=0
A+B=0
B=1
On solving these equations, we obtain
A = −1, C = 1, and B = 1

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Class XII Chapter 7 – Integrals Maths

Hence, the given result is proved.
Question 35:
Integrating by parts, we obtain

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Class XII Chapter 7 – Integrals Maths

Hence, the given result is proved.

Question 36:

Therefore, f (x) is an odd function.
It is known that if f(x) is an odd function, then
Hence, the given result is proved.
Question 37:

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Class XII Chapter 7 – Integrals Maths

Hence, the given result is proved.

Question 38:

Hence, the given result is proved.
Question 39: