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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1.

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

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1

Find an antiderivative (or integral) of the following by the method of inspection:

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.
\({ e }^{ 2x }\)
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:
\(\int { \left( \frac { { x }^{ 3 } }{ { x }^{ 2 } } +\frac { { 5x }^{ 2 } }{ { x }^{ 2 } } -\frac { 4 }{ { x }^{ 2 } } \right) } \)
\(=\int { xdx+5\int { 1dx-4 } \int { { x }^{ 2 }dx } } \)
\(=\frac { { x }^{ 2 } }{ 2 } +5x+\frac { 4 }{ x } +c\)

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) \(f(x)=\int { \left( { 4x }^{ 3 }-\frac { 3 }{ { x }^{ 4 } } \right) dx } \)
\(={ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +c \)
\(\therefore f(2)={ (2) }^{ 4 }+\frac { 1 }{ { (2) }^{ 3 } } +c=0=-\frac { 129 }{ 8 } \)

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