NCERT Solutions For Class 12 Maths Chapter 7 Integrals Ex 7.1
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Textbook | NCERT |
Board | CBSE |
Category | NCERT Solutions |
Class | Class 12 |
Subject | Maths |
Chapter | Chapter 7 |
Exercise | Class 12 Maths Chapter 7 Integrals Exercise 7.1 |
Number of Questions Solved | 22 |

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1
NCERT TEXTBOOK EXERCISES
Ex 7.1 Class 12 Maths Question 1.
sin 2x
Solution:
$\int { sin2x\quad dx=-\frac { cos2x }{ 2 } +C }$
Ex 7.1 Class 12 Maths Question 2.
cos 3x
Solution:
$\int { cos3x\quad dx=\frac { sin3x }{ 3 } +C }$
Ex 7.1 Class 12 Maths Question 3.
e2x
Solution:
$\int { { e }^{ 2x }dx=\frac { { e }^{ 2x } }{ 2 } +C }$
Ex 7.1 Class 12 Maths Question 4.
(ax + c)²
Solution:
$\int { { (ax+b) }^{ 2 }dx=\frac { { (ax+b) }^{ 3 } }{ 3a } } +C$
Ex 7.1 Class 12 Maths Question 5.
${ sin\quad 2x-4e }^{ 3x }$
Solution:
$\int { \left( { sin2x-4e }^{ 3x } \right) dx=-\frac { cos2x }{ 2 } -\frac { { 4e }^{ 3x } }{ 3 } +C }$
Find the following integrals in Exercises 6 to 20 :
Ex 7.1 Class 12 Maths Question 6.
$\int { \left( { 4e }^{ 3x }+1 \right) dx }$
Solution:
$=\int { { 4e }^{ 3x }dx+\int { dx=\frac { 4 }{ 3 } { e }^{ 3x }+x+c } }$
Ex 7.1 Class 12 Maths Question 7.
$\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx }$
Solution:
$=\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx } =\frac { { x }^{ 3 } }{ 3 } -x+C$
Ex 7.1 Class 12 Maths Question 8.
$\int { { (ax }^{ 2 }+bx+c)dx }$
Solution:
$=\frac { { ax }^{ 3 } }{ 3 } +\frac { { bx }^{ 2 } }{ 2 } +cx+d$
Ex 7.1 Class 12 Maths Question 9.
$\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx }$
Solution:
$=\frac { { 2x }^{ 3 } }{ 3 } +{ e }^{ x }+c$
Ex 7.1 Class 12 Maths Question 10.
$\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx }$
Solution:
$=\frac { { x }^{ 2 } }{ 2 } +logx-2x+C$
Ex 7.1 Class 12 Maths Question 11.
$\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx }$
Solution:
$\int { \left( \frac { { x }^{ 3 } }{ { x }^{ 2 } } +\frac { { 5x }^{ 2 } }{ { x }^{ 2 } } -\frac { 4 }{ { x }^{ 2 } } \right) }$
$=\frac { { x }^{ 2 } }{ 2 } +5x+\frac { 4 }{ x } +c$
$=\frac { { x }^{ 2 } }{ 2 } +5x+\frac { 4 }{ x } +c$
Ex 7.1 Class 12 Maths Question 12.
$\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx }$
Solution:
$=\int { \left( { x }^{ \frac { 5 }{ 2 } }+{ 3x }^{ \frac { 1 }{ 2 } }+4{ x }^{ -\frac { 1 }{ 2 } } \right) } dx$
$=\frac { 2 }{ 7 } { x }^{ \frac { 7 }{ 2 } }+{ 2x }^{ \frac { 3 }{ 2 } }+8\sqrt { x } +c$
Ex 7.1 Class 12 Maths Question 13.
$\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx }$
Solution:
$=\int { \frac { { x }^{ 2 }(x-1)+(x-1) }{ x-1 } dx }$
$=\int { \left( { x }^{ 2 }+1 \right) dx } =\frac { { x }^{ 3 } }{ 3 } +x+c$
Ex 7.1 Class 12 Maths Question 14.
$\int { \left( 1-x \right) \sqrt { x } dx }$
Solution:
$=\int { { x }^{ \frac { 1 }{ 2 } }-{ x }^{ \frac { 3 }{ 2 } }dx\quad =\quad \frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }-\frac { 2 }{ 5 } { x }^{ \frac { 5 }{ 2 } } }$
Ex 7.1 Class 12 Maths Question 15.
$\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx }$
Solution:
$=\int { \left( { 3x }^{ \frac { 5 }{ 2 } }+{ 2 }^{ \frac { 3 }{ 2 } }+{ 3x }^{ \frac { 1 }{ 2 } } \right) dx }$
$=\frac { 6 }{ 7 } { x }^{ \frac { 7 }{ 2 } }+\frac { 4 }{ 5 } { x }^{ \frac { 5 }{ 2 } }+\frac { 6 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+c$
Ex 7.1 Class 12 Maths Question 16.
$\int { (2x-3cosx+{ e }^{ x })dx }$
Solution:
$=\frac { { 2x }^{ 2 } }{ 2 } -3sinx+{ e }^{ x }+c$
$={ x }^{ 2 }-3sinx+{ e }^{ x }+c$
Ex 7.1 Class 12 Maths Question 17.
$\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx }$
Solution:
$=\frac { { 2x }^{ 3 } }{ 3 } +3cosx+5\frac { { x }^{ \frac { 3 }{ 2 } } }{ \frac { 3 }{ 2 } } +c$
$=\frac { 2 }{ 3 } { x }^{ 3 }+3cosx+\frac { 10 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+c$
Ex 7.1 Class 12 Maths Question 18.
$\int { secx(secx+tanx)dx }$
Solution:
$=\int { { (sec }^{ 2 }x+secxtanx)dx }$
= tanx + secx + c
Ex 7.1 Class 12 Maths Question 19.
$\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx }$
Solution:
$=\int { \frac { 1 }{ { cos }^{ 2 }x } } { sin }^{ 2 }xdx$
$=\int { tan } ^{ 2 }xdx\quad =\int { { (sec }^{ 2 }x-1)dx\quad =tanx-x+c }$
Ex 7.1 Class 12 Maths 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.
Ex 7.1 Class 12 Maths Question 21.
The antiderivative $\left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right)$ 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) $\int { \left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right) dx }$
$=\int { \left( { x }^{ \frac { 1 }{ 2 } }+{ x }^{ \frac { 1 }{ 2 } } \right) dx }$
$=\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c$
Ex 7.1 Class 12 Maths 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) $f(x)=\int { \left( { 4x }^{ 3 }-\frac { 3 }{ { x }^{ 4 } } \right) dx }$
$={ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +c$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Exercise 7.1 PDF
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Other Chapter of Class 12 Maths Chapter 7 Integrals
- Class 12 Maths Chapter 7 Integrals Ex 7.1
- Class 12 Maths Chapter 7 Integrals Ex 7.2
- Class 12 Maths Chapter 7 Integrals Ex 7.3
- Class 12 Maths Chapter 7 Integrals Ex 7.4
- Class 12 Maths Chapter 7 Integrals Ex 7.5
- Class 12 Maths Chapter 7 Integrals Ex 7.6
- Class 12 Maths Chapter 7 Integrals Ex 7.7
- Class 12 Maths Chapter 7 Integrals Ex 7.8
- Class 12 Maths Chapter 7 Integrals Ex 7.9
- Class 12 Maths Chapter 7 Integrals Ex 7.10
- Class 12 Maths Chapter 7 Integrals Ex 7.11