# NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

These NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1

Question 1.
sin 2x
Solution:

Question 2.
cos 3x
Solution:

Question 3.
$${ e }^{ 2x }$$
Solution:

Question 4.
(ax + c)²
Solution:

Question 5.
sin 2x – 4 e3x
Solution:

∴ An anti derivative of sin2x – 4e3x is – $$\frac { 1 }{ 2 }$$ cos 2x – $$\frac { 4 }{ 3 }$$4e3x

Question 6.
$$\int { \left( { 4e }^{ 3x }+1 \right) dx }$$
Solution:

Question 7.
$$\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx }$$
Solution:
$$=\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) }dx = ∫(x² – 1)dx = ∫x² dx – ∫1 dx = [latex]\frac { { x }^{ 3 } }{ 3 }$$ – x + C

Question 8.
$$\int { { (ax }^{ 2 }+bx+c)dx }$$
Solution:
$$\int { { (ax }^{ 2 }+bx+c)dx }$$
= a∫x² dx + b ∫x dx + c∫1 dx
= $$\frac { { ax }^{ 3 } }{ 3 } +\frac { { bx }^{ 2 } }{ 2 }$$ + cx + C

Question 9.
$$\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx }$$
Solution:
$$\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx }$$
= 2∫x² dx + ∫ex dx
= 2 $$\frac { { x }^{ 3 } }{ 3 }$$ + ex + C
= $$\frac { { 2x }^{ 3 } }{ 3 } +{ e }^{ x }$$ + C

Question 10.
$$\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx }$$
Solution:

Question 11.
$$\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx }$$
Solution:

Question 12.
$$\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx }$$
Solution:

Question 13.
$$\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx }$$
Solution:
$$=\int { \frac { { x }^{ 2 }(x-1)+(x-1) }{ x-1 } dx }$$
$$=\int { \left( { x }^{ 2 }+1 \right) dx } =\frac { { x }^{ 3 } }{ 3 } +x+c$$

Question 14.
$$\int { \left( 1-x \right) \sqrt { x } dx }$$
Solution:

Question 15.
$$\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx }$$
Solution:

Question 16.
$$\int { (2x – 3cosx+{ e }^{ x })dx }$$
Solution:
$$\int { (2x – 3cosx+{ e }^{ x })dx }$$
= 2∫x dx – 3∫cosx dx + ∫ex dx
= 2($$\frac { { x }^{ 2 } }{ 3 }$$) – 3 sin x + ex + C
$$={ x }^{ 2 }-3sinx+{ e }^{ x }$$ + C

Question 17.
$$\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx }$$
Solution:

Question 18.
$$\int { secx(secx+tanx)dx }$$
Solution:
$$\int { secx(secx+tanx)dx }$$
= $$\int\left(\sec ^{2} x+\sec x \tan x\right) d x$$
= ∫sec² x dx + ∫sec x tan x dx
= tan x + sec + C

Question 19.
$$\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx }$$
Solution:
= $$\int { \frac { 1 }{ { cos }^{ 2 }x } } { sin }^{ 2 }xdx$$
= $$\int { tan } ^{ 2 }xdx\quad$$
= $$\int\left(\sec ^{2} x-1\right) d x$$
= tanx – x + c

Question 20.
$$\int { \frac { 2-3sinx }{ { cos }^{ 2 }x } dx }$$
Solution:
= $$\int { \left( \frac { 2 }{ { cos }^{ 2 }x } -3\frac { sinx }{ { cos }^{ 2 }x } \right) dx }$$
= $$\int { ({ 2sec }^{ 2 }x-3secxtanx)dx }$$
= 2tanx – 3secx + c

Choose the correct answer in Exercises 21 and 22.

Question 21.
The anti derivative $$\left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right)$$ equals
(a) $$\frac { 1 }{ 3 } { x }^{ \frac { 1 }{ 3 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$$
(b) $$\frac { 2 }{ 3 } { x }^{ \frac { 2 }{ 3 } }+{ \frac { 1 }{ 2 } x }^{ 2 }+c$$
(c) $$\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$$
(d) $$\frac { 3 }{ 2 } { x }^{ \frac { 3 }{ 2 } }+\frac { 1 }{ 2 } { x }^{ \frac { 1 }{ 2 } }+c$$
Solution:
(c) $$\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$$

Question 22.
If $$\frac { d }{ dx } f(x) = { 4x }^{ 3 } -\frac { 3 }{ { x }^{ 4 } }$$ such that f(2) = 0 then f(x) is
(a) $${ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 }$$
(b) $${ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } +\frac { 129 }{ 8 }$$
(c) $${ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +\frac { 129 }{ 8 }$$
(d) $${ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } -\frac { 129 }{ 8 }$$
Solution:
(a) $${ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 }$$

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