NCERT Solutions | Class 12 Maths Chapter 7

NCERT Solutions | Class 12 Maths Chapter 7 | Integrals 

CBSE Solutions | Maths Class 12

Check the below NCERT Solutions for Class 12 Maths Chapter 7 Integrals Pdf free download. NCERT Solutions Class 12 Maths  were prepared based on the latest exam pattern. We have Provided Integrals Class 12 Maths NCERT Solutions to help students understand the concept very well.

NCERT | Class 12 Maths

NCERT Solutions Class 12 Maths

Book:	National Council of Educational Research and Training (NCERT)
Board:	Central Board of Secondary Education (CBSE)
Class:	12th
Subject:	Maths
Chapter:	7
Chapters Name:	Integrals
Medium:	English

Integrals | Class 12 Maths | NCERT Books Solutions

You can refer to MCQ Questions for Class 12 Maths Chapter 7 Integrals to revise the concepts in the syllabus effectively and improve your chances of securing high marks in your board exams.

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

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2

Integrate the functions in Exercises 1 to 37:

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Ex 7.1 Class 12 Maths Question 1.
\(\frac { 2x }{ 1+{ x }^{ 2 } } \)
Solution.
Let 1+x² = t
⇒ 2xdx = dt
\(\therefore \int { \frac { 2x }{ 1+{ x }^{ 2 } } dx\quad = } \int { \frac { dt }{ t } \quad =logt+C\quad =log(1+{ x }^{ 2 })+C } \)

Ex 7.2 Class 12 Maths Question 2.
\(\frac { { \left( logx \right) }^{ 2 } }{ x } \)
Solution.
Let logx = t
⇒ \(\frac { 1 }{ x }dx=dt\)
\(\therefore \int { \frac { { (logx) }^{ 2 } }{ x } dx } \quad =\int { { t }^{ 2 }dt } \quad =\frac { { t }^{ 3 } }{ 3 } +c\quad =\frac { 1 }{ 3 } { (logx) }^{ 3 }+c\)

Ex 7.2 Class 12 Maths Question 3.
\(\frac { 1 }{ x+xlogx } \)
Solution.
Put 1+logx = t
∴ \(\frac { 1 }{ x }dx=dt\)
\(\int { \frac { 1 }{ x(1+logx) } dx } =\int { \frac { 1 }{ t } dt } =log|t|+c\)
= log|1+logx|+c

Ex 7.2 Class 12 Maths Question 4.
sinx sin(cosx)
Solution.
Put cosx = t, -sinx dx = dt
\(\int { sinx\quad sin(cosx)dx } =-\int { sin(cosx) } (-sinx)dx\)
\(=-\int { sint\quad dt } \quad =cost+c\quad =cos(cosx)+c\)

Ex 7.2 Class 12 Maths Question 5.
sin(ax+b) cos(ax+b)
Solution.
let sin(ax+b) = t
⇒ cos(ax+b)dx = dt
\(\therefore \int { sin(ax+b)cos(ax+b)dx } =\frac { 1 }{ a } \int { t\quad dt } \)
\(=\frac { 1 }{ a } .\frac { { t }^{ 2 } }{ 2 } +c\quad =\frac { 1 }{ 2a } { sin }^{ 2 }(ax+b)+C\)

Ex 7.2 Class 12 Maths Question 6.
\(\sqrt { ax+b } \)
Solution.
\(\int { \sqrt { ax+b } dx } \quad =\frac { 2 }{ 3a } { (ax+b) }^{ \frac { 3 }{ 2 } }+C\)

Ex 7.2 Class 12 Maths Question 7.
\(x\sqrt { x+2 } \)
Solution.
Let x+2 = t²
⇒ dx = 2t dt

Ex 7.2 Class 12 Maths Question 8.
\(x\sqrt { 1+{ 2x }^{ 2 } } \)
Solution.
let 1+2x² = t²
⇒ 4x dx = 2t dt
\(x\quad dx=\frac { t }{ 2 } dt\int { x\sqrt { 1+{ 2x }^{ 2 } } dx } \)
\(=\frac { 1 }{ 2 } \int { { t }^{ 2 }dt } =\frac { { t }^{ 3 } }{ 6 } +c=\frac { 1 }{ 6 } { ({ 1+2x }^{ 2 }) }^{ \frac { 3 }{ 2 } }+c\)

Ex 7.2 Class 12 Maths Question 9.
\((4x+2)\sqrt { { x }^{ 2 }+x+1 } \)
Solution.
let x²+x+1 = t
⇒(2x+1)dx = dt
\(\therefore \int { (4x+1)\sqrt { { x }^{ 2 }+x+1 } dx } =2\int { \sqrt { t } dt } \)
\(=\frac { { 2t }^{ \frac { 3 }{ 2 } } }{ ^{ \frac { 3 }{ 2 } } } +c\quad =\frac { 4 }{ 3 } { t }^{ \frac { 3 }{ 2 } }+c\quad =\frac { 4 }{ 3 } { ({ x }^{ 2 }+x+1) }^{ \frac { 3 }{ 2 } }+c\)

Ex 7.2 Class 12 Maths Question 10.
\(\frac { 1 }{ x-\sqrt { x } } \)
Solution.
\(\int { \frac { 1 }{ x-\sqrt { x } } dx } =\int { \frac { 1 }{ \sqrt { x } (\sqrt { x-1 } ) } dx } =I\)
Let √x-1 = t
\(\frac { 1 }{ 2 } { x }^{ -\frac { 1 }{ 2 } }dx=dt\)
\(I=2\int { \frac { dt }{ t } } \)
= 2logt + c
= 2log(√x-1)+c

Ex 7.2 Class 12 Maths Question 11.
\(\frac { x }{ \sqrt { x+4 } } ,x>0\)
Solution.
let x+4 = t
⇒ dx = dt, x = t-4

Ex 7.2 Class 12 Maths Question 12.
\({ { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }\)
Solution.
\(\int { { { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }.dx } \quad =\frac { 1 }{ 7 } { { (x }^{ 3 }-1) }^{ \frac { 7 }{ 3 } }+\frac { 1 }{ 4 } { { (x }^{ 3 }-1) }^{ \frac { 4 }{ 3 } }+c\)

Ex 7.2 Class 12 Maths Question 13.
\(\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } } \)
Solution.
Let 2+3x³ = t
⇒ 9x² dx = dt
\(\therefore \int { \frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } } dx } =\frac { 1 }{ 9 } \int { \frac { dt }{ { t }^{ 3 } } =\frac { 1 }{ 9 } \int { { t }^{ -3 }dt } } \)
\(=-\frac { 1 }{ { 18t }^{ 2 } } +c\quad =-\frac { 1 }{ 18(2+{ 3x }^{ 3 })^{ 2 } } +c\)

Ex 7.2 Class 12 Maths Question 14.
\(\frac { 1 }{ x(logx)^{ m } } ,x>0\)
Solution.
Put log x = t, so that \(\frac { 1 }{ x }dx=dt\)
\(\therefore \int { \frac { 1 }{ { x(logx) }^{ m } } dx } =\int { \frac { dt }{ { t }^{ m } } =\frac { { t }^{ -m+1 } }{ -m+1 } +c } \)
\(=\frac { { (logx) }^{ 1-m } }{ 1-m } +c\)

Ex 7.2 Class 12 Maths Question 15.
\(\frac { x }{ 9-4{ x }^{ 2 } } \)
Solution.
put 9-4x² = t, so that -8x dx = dt
\(\therefore \int { \frac { x }{ 9-{ 4x }^{ 2 } } dx } =-\frac { 1 }{ 8 } \int { \frac { dt }{ t } } =-\frac { 1 }{ 8 } log|t|+c\)
\(=\frac { 1 }{ 8 } log\frac { 1 }{ |9-{ 4x }^{ 2 }| } +c\)

Ex 7.2 Class 12 Maths Question 16.
\({ e }^{ 2x+3 }\)
Solution.
put 2x+3 = t
so that 2dx = dt
\(\int { { e }^{ 2x+3 } } dx\quad =\frac { 1 }{ 2 } \int { { e }^{ t }dt } \quad =\frac { 1 }{ 2 } { e }^{ t }+c\quad =\frac { 1 }{ 2 } { e }^{ 2x+3 }+c\)

Ex 7.2 Class 12 Maths Question 17.
\(\frac { x }{ { e }^{ { x }^{ 2 } } } \)
Solution.
Let x² = t
⇒ 2xdx = dt ⇒ \(xdx=\frac { dt }{ 2 }\)
\(\therefore \int { \frac { x }{ { e }^{ { x }^{ 2 } } } dx } \quad =\frac { 1 }{ 2 } \int { \frac { dt }{ { e }^{ t } } \quad =\frac { 1 }{ 2 } \int { { e }^{ -t } } dt } \)
\(=-\frac { 1 }{ 2 } { e }^{ { -x }^{ 2 } }+c \)

Ex 7.2 Class 12 Maths Question 18.
\(\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } \)
Solution.
\(let\quad { tan }^{ -1 }x=t\Rightarrow \frac { 1 }{ 1+{ x }^{ 2 } } dx=dt\)
\(\therefore \int { \frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } dx } \quad =\int { { e }^{ t }dt\quad ={ e }^{ { tan }^{ -1 }x }+c } \)

Ex 7.2 Class 12 Maths Question 19.
\(\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 } \)
Solution.
\(\int { \frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 } dx\quad =\int { \frac { { e }^{ x }({ e }^{ x }-{ e }^{ -x }) }{ { e }^{ x }({ e }^{ x }+{ e }^{ -x }) } dx=I } } \)
put e^x+e^-x = t
so that (e^x-e^-x)dx = dt
\(\therefore I=\int { \frac { dt }{ t } =log|t|+c } =log|{ e }^{ x }+{ e }^{ -x }|+c\)

Ex 7.2 Class 12 Maths Question 20.
\(\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } } \)
Solution.
put e^2x-e^-2x = t
so that (2e^2x-2e^-2x)dx = dt
\(\therefore \int { \frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } } } dx=\frac { 1 }{ 2 } \int { \frac { 1 }{ t } dt } =\frac { 1 }{ 2 } log|t|+c\)
\(=\frac { 1 }{ 2 } log+|{ e }^{ 2x }+{ e }^{ -2x }|+c\)

Ex 7.2 Class 12 Maths Question 21.
tan²(2x-3)
Solution.
∫tan²(2x-3)dx = ∫[sec²(2x-3)-1]dx = I
put 2x-3 = t
so that 2dx = dt
I = \(\frac { 1 }{ 2 }\) ∫sec²t dt-x+c
= \(\frac { 1 }{ 2 }t-x+c\)
= \(\frac { 1 }{ 2 }tan(2x-3)-x+c\)

Ex 7.2 Class 12 Maths Question 22.
sec²(7-4x)
Solution.
∫sec²(7-4x)dx
= \(\frac { tan(7-4x) }{ -4 }+c\)

Ex 7.2 Class 12 Maths Question 23.
\(\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } \)
Solution.
\(let\quad { sin }^{ -1 }x=t\quad \Rightarrow \frac { 1dx }{ \sqrt { 1-{ x }^{ 2 } } } =dt\)
\(\int { \frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx } =\int { t\quad dt } =\frac { 1 }{ 2 } { t }^{ 2 }+c=\frac { 1 }{ 2 } { { (sin }^{ -1 }x) }^{ 2 }+c\)

Ex 7.2 Class 12 Maths Question 24.
\(\frac { 2cosx-3sinx }{ 6cosx+4sinx }\)
Solution.
put 2sinx+4cosx = t
⇒ (2cosx-3sinx)dx = dt
\(\frac { 1 }{ 2 } \int { \frac { 2cosx-3sinx }{ 2sinx+3cosx } dx } =\frac { 1 }{ 2 } \int { \frac { dt }{ t } } =\frac { 1 }{ 2 } log|t|+c\)
\(\frac { 1 }{ 2 } log|2sinx+3cosx|+c\)

Ex 7.2 Class 12 Maths Question 25.
\(\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } } \)
Solution.
put 1-tanx = t
so that -sec²x dx = dt
\(\therefore \int { \frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } } dx } =\int { \frac { { sec }^{ 2 }x }{ { (1-tanx) }^{ 2 } } dx } \)
\(=-\int { \frac { dt }{ { t }^{ 2 } } } =\frac { 1 }{ t } +c=\frac { 1 }{ (1-tanx) } +c \)

Ex 7.2 Class 12 Maths Question 26.
\(\frac { cos\sqrt { x } }{ \sqrt { x } } \)
Solution.
\(put\quad \sqrt { x } =t,so\quad that\frac { 1 }{ 2\sqrt { x } } dx=dt\)
\(\therefore \int { \frac { cos\sqrt { x } }{ \sqrt { x } } } dx\quad =\quad 2\quad =\int { cost\quad dt\quad = } 2sint+c\)
= 2sin√x+c

Ex 7.2 Class 12 Maths Question 27.
\(\sqrt { sin2x } cos2x\)
Solution.
put sin2x = t²
⇒ cos2x dx = t dt
\(\therefore \int { \sqrt { sin2x } .cos2x\quad dx } \quad =\int { t.tdt=\frac { { t }^{ 3 } }{ 3 } +c } \)
\(=\frac { { (sin2x) }^{ \frac { 3 }{ 2 } } }{ 3 } +c \)

Ex 7.2 Class 12 Maths Question 28.
\(\frac { cosx }{ \sqrt { 1+sinx } } \)
Solution.
put 1+sinx = t²
⇒cosx dx = 2t dt
\(\therefore \int { \frac { cosx }{ \sqrt { 1+sinx } } dx } =2\int { dt } =2t+c\)
\(=2\sqrt { 1+sinx } +c\)

Ex 7.2 Class 12 Maths Question 29.
cotx log sinx
Solution.
put log sinx = t,
⇒ cot x dx = dt
\(\therefore \int { cot\quad logsinx\quad dx } =\int { t } dt\quad =\frac { { t }^{ 2 } }{ 2 } +c\)
\(=\frac { 1 }{ 2 } { (log\quad sinx) }^{ 2 }+c\)

Ex 7.2 Class 12 Maths Question 30.
\(\frac { sinx }{ 1+cosx }\)
Solution.
put 1+cosx = t
⇒ -sinx dx = dt
\(\therefore \int { \frac { sinx }{ 1+cosx } dx } =\int { -\frac { dt }{ t } } =-logt+c\)
=-log(1+cosx)+c

Ex 7.2 Class 12 Maths Question 31.
\(\frac { sinx }{ { (1+cosx) }^{ 2 } } \)
Solution.
put 1+cosx = t
so that -sinx dx = dt
\(\therefore \int { \frac { sinx }{ { (1+cosx) }^{ 2 } } dx } =-\int { \frac { dt }{ { t }^{ 2 } } } \)
\(=\frac { 1 }{ t } +c=\frac { 1 }{ 1+cosx } +c \)

Ex 7.2 Class 12 Maths Question 32.
\(\frac { 1 }{ 1+cotx }\)
Solution.
\(\int { \frac { 1 }{ 1+\frac { cosx }{ sinx } } } dx=\frac { 1 }{ 2 } \int { \frac { 2sinx\quad dx }{ sinx+cosx } } \)

Ex 7.2 Class 12 Maths Question 33.
\(\frac { 1 }{ 1-tanx }\)
Solution.
\(\int { \frac { 1 }{ 1-tanx } } dx=\frac { 1 }{ 2 } \int { \frac { 2cosx\quad dx }{ cosx-sinx } } \)

Ex 7.2 Class 12 Maths Question 34.
\(\frac { \sqrt { tanx } }{ sinxcosx } \)
Solution.
\(\int { \frac { \sqrt { tanx } }{ sinxcosx } dx } =\int { \frac { \sqrt { tanx } }{ tanx } } .{ sec }^{ 2 }xdx\)

Ex 7.2 Class 12 Maths Question 35.
\(\frac { { (1+logx) }^{ 2 } }{ x } \)
Solution.
let 1+logx = t
⇒ \(\frac { 1 }{ x }dx=dt\)
\(\int { \frac { { (1+logx) }^{ 2 } }{ x } dx } =\int { { t }^{ 2 }dt } =\frac { { t }^{ 3 } }{ 3 } +c\)
\(=\frac { 1 }{ 3 } { (1+logx) }^{ 3 }+c\)

Ex 7.2 Class 12 Maths Question 36.
\(\frac { (x+1){ (x+logx) }^{ 2 } }{ x } \)
Solution.
put x+logx = t
\(\left( \frac { x+1 }{ x } \right) dx=dt\)
\(\therefore \int { \frac { { (x+1)(x+logx) }^{ 2 } }{ x } } dx=\int { { t }^{ 2 }dt } \)
\(=\frac { { (x+logx) }^{ 3 } }{ 3 } +c\)

Ex 7.2 Class 12 Maths Question 37.
\(\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx \)
Solution.
\(put\quad { tan }^{ -1 }{ x }^{ 4 }=t\quad so\quad that\frac { 1 }{ 1+{ x }^{ 8 } } .{ 4x }^{ 3 }dx=dt\)
\(\therefore \int { \frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } } dx=\frac { 1 }{ 4 } \int { sint\quad dt } \)
\(=\frac { 1 }{ 4 } (-cost)+c=-\frac { 1 }{ 4 } cos({ tan }^{ -1 }{ x }^{ 4 })+c\)

Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.
\(\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx } \)
(a) 10^x – x¹⁰ + C
(b) 10^x + x¹⁰ + C
(c) (10^x – x¹⁰) + C
(d) log (10^x + x¹⁰) + C
Solution.
(d) \(\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx } \)
= log (10^x + x¹⁰) + C

Ex 7.2 Class 12 Maths Question 39.
\(\int { \frac { dx }{ { sin }^{ 2 }x{ \quad cos }^{ 2 }x } = } \)
(a) tanx + cotx + c
(b) tanx – cotx + c
(c) tanx cotx + c
(d) tanx – cot2x + c
Solution.
(c) \(\int { \frac { dx }{ { sin }^{ 2 }x{ \quad cos }^{ 2 }x } = } \)
\(=\int { \left( { sec }^{ 2 }x+{ cosec }^{ 2 }x \right) dx } \)
= tanx – cotx + c

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

Find the integrals of the functions in Exercises 1 to 22.

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Ex 7.3 Class 12 Maths Question 1.
sin²(2x+5)
Solution:
∫sin²(2x+5)dx
= \(\frac { 1 }{ 2 }\)∫[1-cos2(2x+5)]dx
= \(\frac { 1 }{ 2 }\)∫[1-cos(4x+10)]dx
= \(\frac { 1 }{ 2 } \left[ x-\frac { sin(4x+10) }{ 4 } \right] +c\)

Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution.
∫sin3x cos4x
= \(\frac { 1 }{ 2 }\)∫[sin(3x+4x)+cos(3x-4x)]dx
= \(\frac { 1 }{ 2 }\)∫[sin7x+sin(-x)]dx
= \(-\frac { 1 }{ 14 } cos7x+\frac { 1 }{ 2 } cosx+c\)

Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution.
\(\frac { 1 }{ 2 }\) ∫cos2x cos4x cos6x dx
= \(\frac { 1 }{ 2 }\) ∫(cos6x+cos2x) cos6x dx

Ex 7.3 Class 12 Maths Question 4.
∫sin³(2x+1)dx
Solution.
= \(\frac { 1 }{ 4 }\) ∫[3sin(2x+1)-sin3(2x+1)]dx
= \(-\frac { 3 }{ 8 } cos(2x+1)+\frac { 1 }{ 24 } [4{ cos }^{ 3 }(2x+1)-3cos(2x+1)]+c\)
= \(-\frac { 1 }{ 2 } cos(2x+1)+\frac { 1 }{ 6 } { cos }^{ 3 }(2x+1)+c\)

Ex 7.3 Class 12 Maths Question 5.
sin³x cos³x
Solution.
put sin x = t
⇒ cos x dx = dt
\(\therefore \int { { sin }^{ 3 }x{ cos }^{ 3 }xdx } =\int { { t }^{ 3 }(1-{ t }^{ 2 })dt } \)
\(\frac { { t }^{ 4 } }{ 4 } -\frac { { t }^{ 6 } }{ 6 } +c=\frac { { (sinx) }^{ 4 } }{ 4 } -\frac { { (sinx) }^{ 6 } }{ 6 } +c\)

Ex 7.3 Class 12 Maths Question 6.
sinx sin2x sin3x
Solution.
∫sinx sin2x sin3x dx
= \(\frac { 1 }{ 2 }\) ∫ 2sin x sin 2x sin 3x dx
= \(\frac { 1 }{ 2 }\) ∫ (cosx – cos3x)sin 3x dx
= \(\frac { 1 }{ 2 }\) ∫ (sin 4x + sin 2x – sin 6x)dx
= \(\frac { 1 }{ 4 } \left\{ \frac { -cos4x }{ 4 } -\frac { cos2x }{ 2 } +\frac { cos6x }{ 6 } \right\} +c\)

Ex 7.3 Class 12 Maths Question 7.
sin 4x sin 8x
Solution.
\(\frac { 1 }{ 2 }\)∫sin 4x sin 8xdx
= \(\frac { 1 }{ 2 }\)∫(cos 4x – cos 12x)dx
= \(\frac { 1 }{ 2 } \left[ \frac { sin4x }{ 4 } -\frac { sin12x }{ 12 } \right] +c\)

Ex 7.3 Class 12 Maths Question 8.
\(\frac { 1-cosx }{ 1+cosx }\)
Solution.
\(\int { \frac { 1-cosx }{ 1+cosx } dx } \)
\(\int { \frac { { 2sin }^{ 2 }\frac { x }{ 2 } }{ { 2cos }^{ 2 }\frac { x }{ 2 } } dx } =\int { { tan }^{ 2 }\frac { x }{ 2 } dx } \)
\(=\int { \left[ { sec }^{ 2 }\frac { x }{ 2 } -1 \right] } dx\quad =2tan\frac { x }{ 2 } -x+c \)

Ex 7.3 Class 12 Maths Question 9.
\(\frac { cosx }{ 1+cosx }\)
Solution.
\(\int { \frac { cosx }{ 1+cosx } dx } \)
\(=\int { 1 } dx-\int { \frac { 1 }{ 1+cosx } dx } \)
\(=x-\frac { 1 }{ 2 } \int { { sec }^{ 2 }\frac { x }{ 2 } dx+c\quad =x-tan\frac { x }{ 2 } +c } \)

Ex 7.3 Class 12 Maths Question 10.
∫sinx⁴ dx
Solution.
\(\int { { (\frac { 1-cos2x }{ 2 } ) }^{ 2 }dx } \quad =\frac { 1 }{ 4 } \int { \left( { 1+cos }^{ 2 }2x-2cos2x \right) dx } \)

Ex 7.3 Class 12 Maths Question 11.
cos⁴ 2x
Solution.
∫ cos⁴ 2x dx
\(\int { { \left( \frac { 1+cos4x }{ 2 } \right) }^{ 2 } } dx\)

Ex 7.3 Class 12 Maths Question 12.
\(\frac { { sin }^{ 2 }x }{ 1+cosx } \)
Solution.
\(\int { \frac { { sin }^{ 2 }x }{ 1+cosx } } dx\quad =\int { \frac { 1-{ cos }^{ 2 }x }{ 1+cosx } } dx\)
\(\int { (1-cosx) } dx\quad =x-sinx+c\)

Ex 7.3 Class 12 Maths Question 13.
\(\frac { cos2x-cos2\alpha }{ cosx-cos\alpha } \)
Solution.
let I = \(\int { \frac { \left( { 2cos }^{ 2 }x-1 \right) -\left( { 2cos }^{ 2 }\alpha -1 \right) }{ cosx-cos\alpha } } dx\)
\(\int { \frac { 2\left( { cos }x-cos\alpha \right) -\left( { cos }x+cos\alpha \right) }{ cosx-cos\alpha } } dx\)
= 2∫cos x dx + 2cos α∫dx
= 2(sinx+xcosα)+c

Ex 7.3 Class 12 Maths Question 14.
\(\frac { cosx-sinx }{ 1+sin2x }\)
Solution.
let I = \(\int { \frac { cosx-sinx }{ 1+sin2x } } dx=\int { \frac { cosx-sinx }{ { (cosx+sinx) }^{ 2 } } dx } \)
put cosx+sinx = t
⇒ (-sinx+cosx)dx = dt
\(I=\int { \frac { dt }{ { t }^{ 2 } } } =-\frac { 1 }{ t } +c\quad =\frac { -1 }{ cosx+sinx } +c \)

Ex 7.3 Class 12 Maths Question 15.
\(\int { { tan }^{ 3 }2x\quad sec2x\quad dx=I } \)
Solution.
I = ∫(sec²2x-1)sec2x tan 2xdx
put sec2x=t,2 sec2x tan2x dx=dt

Ex 7.3 Class 12 Maths Question 16.
tan⁴x
Solution.
let I = ∫tan⁴ dx
= ∫(sec²x-1)²dx

Ex 7.3 Class 12 Maths Question 17.
\(\frac { { sin }^{ 3 }x+{ cos }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } \)
Solution.
\(\int { \left( \frac { { sin }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } +\frac { { cos }^{ 2 }x }{ sinx{ cos }^{ 2 }x } \right) dx } \)
= secx-cosecx+c

Ex 7.3 Class 12 Maths Question 18.
\(\frac { cos2x+{ 2sin }^{ 2 }x }{ { cos }^{ 2 }x } \)
Solution.
\(I=\int { \frac { \left( { cos }^{ 2 }x-{ sin }^{ 2 }x \right) +2{ sin }^{ 2 }x }{ { cos }^{ 2 }x } } dx\)
\(=\int { \frac { \left( { cos }^{ 2 }x-{ sin }^{ 2 }x \right) }{ { cos }^{ 2 }x } } dx\quad =\int { { sec }^{ 2 }xdx\quad =tanx+c } \)

Ex 7.3 Class 12 Maths Question 19.
\(\frac { 1 }{ sinx{ cos }^{ 3 }x } \)
Solution.
\(I=\int { \left( tanx+\frac { 1 }{ tanx } \right) } { sec }^{ 2 }xdx\)
put tanx = t
so that sec²x dx = dt
\(I=\int { \left( t+\frac { 1 }{ t } \right) } dt\quad =\frac { { t }^{ 2 } }{ 2 } +log|t|+c\)
\(=log|tanx|+\frac { 1 }{ 2 } { tan }^{ 2 }x+c\)

Ex 7.3 Class 12 Maths Question 20.
\(\frac { cos2x }{ { (cosx+sinx) }^{ 2 } } \)
Solution.
\(I=\int { \frac { { cos }^{ 2 }x-{ sin }^{ 2 }x }{ (cosx+sinx)^{ 2 } } dx } =\int { \frac { cosx-sinx }{ cosx+sinx } dx } \)
put cosx+sinx=t
⇒(-sinx+cox)dx = dt
\(I=\int { \frac { dt }{ t } } =log|t|+c\quad =log|cosx+sinx|+c \)

Ex 7.3 Class 12 Maths Question 21.
sin^-1 (cos x)
Solution.
\(\int { { sin }^{ -1 }(cosx)dx } \quad ={ sin }^{ -1 }\left[ sin\left( \frac { \pi }{ 2 } -x \right) \right] dx\)
\(\int { \left( \frac { \pi }{ 2 } -x \right) dx } \quad =\frac { \pi x }{ 2 } -\frac { { x }^{ 2 } }{ 2 } +c\)

Ex 7.3 Class 12 Maths Question 22.
\(\int { \frac { 1 }{ cos(x-a)cos(x-b) } dx } \)
Solution.
\(\frac { 1 }{ sin(a-b) } \int { \frac { sin[(x-b)-(x-a)] }{ cos(x-a)cos(x-b) } dx } \)
\(=\frac { 1 }{ sin(a-b) } \left[ \int { tan(x-b)dx-\int { tan(x-a)dx } } \right] \)
\(=\frac { 1 }{ sin(a-b) } log\left| \frac { cos(x-a) }{ cos(x-b) } \right| +c \)

Ex 7.3 Class 12 Maths Question 23.
\(\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\quad is\quad equal\quad to\)
(a) tanx+cotx+c
(b) tanx+cosecx+c
(c) -tanx+cotx+c
(d) tanx+secx+c
Solution.
(a) \(\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\)
= ∫(sec²x-cosec²x)dx
= tanx+cotx+c

Ex 7.3 Class 12 Maths Question 24.
\(\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\quad is\quad equal\quad to\)
(a) -cot(e.x^x)+c
(b) tan(xe^x)+c
(c) tan(e^x)+c
(d) cot e^x+c
Solution.
(b) \(\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\)
= ∫sec²t dt
= tan t+c = tan(xe^x)+c

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

Integrate the functions in exercises 1 to 23

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Ex 7.4 Class 12 Maths Question 1.
\(\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } \)
Solution.
Let x³ = t ⇒ 3x²dx = dt
\(\int { \frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } dx } =\int { \frac { dt }{ { t }^{ 2 }+1 } } ={ tan }^{ -1 }t+c\)
= tan^-1 (x³)+c

Ex 7.4 Class 12 Maths Question 2.
\(\frac { 1 }{ \sqrt { 1+{ 4x }^{ 2 } } } \)
Solution.
\(\frac { 1 }{ 2 } \int { \frac { dx }{ \sqrt { \frac { 1 }{ 4 } +{ x }^{ 2 } } } } =\frac { 1 }{ 2 } \int { \frac { dx }{ \sqrt { { \left( \frac { 1 }{ 2 } \right) }^{ 2 }+{ x }^{ 2 } } } } \)
\(=\frac { 1 }{ 2 } log\left| 2x+\sqrt { 1+{ 4x }^{ 2 } } \right| +c \)

Ex 7.4 Class 12 Maths Question 3.
\(\frac { 1 }{ \sqrt { { (2-x) }^{ 2 }+1 } } \)
Solution.
put (2-x)=t
so that -dx=dt
⇒ dx=-dt
\(\int { \frac { dx }{ \sqrt { { (2-x) }^{ 2 }+1 } } } =-\int { \frac { dt }{ \sqrt { { t }^{ 2 }+1 } } } =-log|t+\sqrt { { t }^{ 2 }+1 } |+c\)
\(=log\left| \frac { 1 }{ (2-x)+\sqrt { { x }^{ 2 }-4x+5 } } \right| +c\)

Ex 7.4 Class 12 Maths Question 4.
\(\frac { 1 }{ \sqrt { 9-{ 25x }^{ 2 } } } \)
Solution.
\(\int { \frac { dx }{ \sqrt { 9-{ 25x }^{ 2 } } } } =\frac { 1 }{ 5 } \int { \frac { dx }{ \sqrt { { \left( \frac { 3 }{ 5 } \right) }^{ 2 }-{ x }^{ 2 } } } } \)
\(=\frac { 1 }{ 5 } { sin }^{ -1 }\left( \frac { x }{ \frac { 3 }{ 5 } } \right) +c\quad =\frac { 1 }{ 5 } { sin }^{ -1 }\left( \frac { 5x }{ 3 } \right) +c \)

Ex 7.4 Class 12 Maths Question 5.
\(\frac { 3x }{ 1+{ 2x }^{ 4 } } \)
Solution.
Put x²=t,so that 2x dx=dt
⇒x dx = \(\frac { dt }{ 2 }\)
\(\therefore \int { \frac { 3x }{ 1+{ 2x }^{ 4 } } dx } =\frac { 1 }{ 2 } \int { \frac { dt }{ 1+{ 2t }^{ 2 } } } =\frac { 3 }{ 4 } \int { \frac { dt }{ { \left( \frac { 1 }{ \sqrt { 2 } } \right) }^{ 2 }+{ t }^{ 2 } } } \)
\(=\frac { 3 }{ 2\sqrt { 2 } } { tan }^{ -1 }(\sqrt { 2t } )+c\quad =\frac { 3 }{ 2\sqrt { 2 } } { tan }^{ -1 }(\sqrt { { 2x }^{ 2 } } )+c \)

Ex 7.4 Class 12 Maths Question 6.
\(\frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } } \)
Solution.
put x³ = t,so that 3x²dx = dt
\(\int { \frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } } dx } \quad =\frac { 1 }{ 3 } \int { \frac { dt }{ 1-{ t }^{ 2 } } \quad =\frac { 1 }{ 6 } log } \left| \frac { 1+t }{ 1-t } \right| +c\)
\(=\frac { 1 }{ 6 } log\left| \frac { 1+{ x }^{ 3 } }{ 1-{ x }^{ 3 } } \right| +c\)

Ex 7.4 Class 12 Maths Question 7.
\(\frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } } \)
Solution.
\(I=\int { \frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } } dx } -\int { \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } dx } ,I={ I }_{ 1 }-{ I }_{ 2 }\)
put x²-1 = t,so that 2x dx = dt

Ex 7.4 Class 12 Maths Question 8.
\(\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }+{ a }^{ 6 } } } \)
Solution.
put x³ = t
so that 3x²dx = dt
\(I=\frac { 1 }{ 3 } \int { \frac { dt }{ { t }^{ 2 }+{ { (a }^{ 3 }) }^{ 2 } } =\frac { 1 }{ 3 } log\left| t+\sqrt { { t }^{ 2 }+{ a }^{ 6 } } \right| +c } \)
\(=\frac { 1 }{ 3 } log|{ x }^{ 3 }+\sqrt { { a }^{ 6 }+{ x }^{ 6 } } |+c \)

Ex 7.4 Class 12 Maths Question 9.
\(\frac { { sec }^{ 2 }x }{ \sqrt { { tan }^{ 2 }x+4 } } \)
Solution.
let tanx = t
sec x²dx = dt
\(I=\int { \frac { dt }{ \sqrt { { t }^{ 2 }+{ (2) }^{ 2 } } } } =log|t+\sqrt { { t }^{ 2 }+4 } |+c\)
\(=log|tanx+\sqrt { { tan }^{ 2 }x+4 } |+c\)

Ex 7.4 Class 12 Maths Question 10.
\(\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } } \)
Solution.
\(\int { \frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } } dx } =\int { \frac { dx }{ \sqrt { { (x+1) }^{ 2 }+1 } } } \)
\(=log|(x+1)+\sqrt { { x }^{ 2 }+2x+2 } |+c \)

Ex 7.4 Class 12 Maths Question 11.
\(\frac { 1 }{ { 9x }^{ 2 }+6x+5 } \)
Solution.
\(\int { \frac { 1 }{ { 9x }^{ 2 }+6x+5 } } =\frac { 1 }{ 9 } \int { \frac { dx }{ { \left( x+\frac { 1 }{ 3 } \right) }^{ 2 }{ +\left( \frac { 2 }{ 3 } \right) }^{ 2 } } } \)
\(=\frac { 1 }{ 6 } { tan }^{ -1 }\left( \frac { 3x+1 }{ 2 } \right) +c\)

Ex 7.4 Class 12 Maths Question 12.
\(\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } } \)
Solution.
\(I=\int { \frac { dx }{ \sqrt { { 4 }^{ 2 }-{ (x+3) }^{ 2 } } } } \quad ={ sin }^{ -1 }\left( \frac { x+3 }{ 4 } \right) +c\)

Ex 7.4 Class 12 Maths Question 13.
\(\frac { 1 }{ \sqrt { (x-1)(x-2) } } \)
Solution.
\(\int { \frac { 1 }{ \sqrt { (x-1)(x-2) } } dx } =\int { \frac { dx }{ \sqrt { { \left( x-\frac { 3 }{ 2 } \right) }^{ 2 }-{ \left( \frac { 1 }{ 2 } \right) }^{ 2 } } } } \)
\(=log\left| x-\frac { 3 }{ 2 } +\sqrt { { x }^{ 2 }-3x+2 } \right| +c\)

Ex 7.4 Class 12 Maths Question 14.
\(\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } } \)
Solution.
\(\int { \frac { dx }{ \sqrt { 8+3x-{ x }^{ 2 } } } } =\int { \frac { dx }{ \sqrt { 8-\left( { x }^{ 2 }-3x \right) } } } \)
\(=\int { \frac { dx }{ \sqrt { { \left( \frac { \sqrt { 41 } }{ 2 } \right) }^{ 2 }-{ \left( x-\frac { 3 }{ 2 } \right) }^{ 2 } } } } \quad ={ sin }^{ -1 }\left( \frac { 2x-3 }{ \sqrt { 41 } } \right) +c \)

Ex 7.4 Class 12 Maths Question 15.
\(\frac { 1 }{ \sqrt { (x-a)(x-b) } } \)
Solution.
\(\int { \frac { dx }{ \sqrt { (x-a)(x-b) } } } =\int { \frac { dx }{ { \left( x-\frac { a+b }{ 2 } \right) }^{ 2 }-{ \left( \frac { a-b }{ 2 } \right) }^{ 2 } } } \)
\(=log\left| \left( x-\frac { a+b }{ 2 } \right) +\sqrt { (x-a)(x-b) } \right| +c \)

Ex 7.4 Class 12 Maths Question 16.
\(\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } \)
Solution.
\(let\quad I=\int { \frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } } dx\)
put 2x²+x-3=t
so that (4x+1)dx=dt
\(let\quad I=\int { \frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } } dx\)
\(\therefore I=\int { \frac { dt }{ \sqrt { t } } } ={ 2t }^{ \frac { 1 }{ 2 } }+c\quad =2\sqrt { { 2x }^{ 2 }+x-3 } +c\)

Ex 7.4 Class 12 Maths Question 17.
\(\frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } } \)
Solution.
\(\int { \frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } } dx } \quad =\int { \frac { x }{ \sqrt { { x }^{ 2 }-1 } } dx } +\int { \frac { 2 }{ \sqrt { { x }^{ 2 }-1 } } dx } \)

Ex 7.4 Class 12 Maths Question 18.
\(\frac { 5x-2 }{ 1+2x+{ 3x }^{ 2 } } \)
Solution.
put 5x-2=A\(\frac { d }{ dx }\)(1+2x+3x²)+B
⇒ 6A=5, A=\(\frac { 5 }{ 6 }-2=2A+B\), B=\(-\frac { 11 }{ 3 }\)

Ex 7.4 Class 12 Maths Question 19.
\(\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } } \)
Solution.
\(\int { \frac { 6x+7 }{ \sqrt { (x-5)(x-4) } } dx } =\int { \frac { (6x+7)dx }{ \sqrt { { x }^{ 2 }-9x+20 } } } \)

Ex 7.4 Class 12 Maths Question 20.
\(\frac { x+2 }{ \sqrt { 4x-{ x }^{ 2 } } } \)
Solution.
\(I=\int { \frac { x-2 }{ \sqrt { 4-{ (x-2) }^{ 2 } } } dx+4\int { \frac { dx }{ \sqrt { 4-{ (x-2) }^{ 2 } } } } } \)

Ex 7.4 Class 12 Maths Question 21.
\(\frac { x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } } \)
Solution.
\(I=\frac { 1 }{ 2 } \int { \frac { 2x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } } dx } \)

Ex 7.4 Class 12 Maths Question 22.
\(\frac { x+3 }{ { x }^{ 2 }-2x-5 } \)
Solution.
\(I=\frac { 1 }{ 2 } \int { \frac { 2x-2 }{ { x }^{ 2 }-2x-5 } dx } +\int { \frac { dx }{ { x }^{ 2 }-2x-5 } } \)

Ex 7.4 Class 12 Maths Question 23.
\(\frac { 5x+3 }{ \sqrt { { x }^{ 2 }+4x+10 } } \)
Solution.
\(I=\int { \frac { \frac { 5 }{ 2 } (2x+4)+(3-10) }{ \sqrt { { x }^{ 2 }+4x+10 } } dx } \)

Ex 7.4 Class 12 Maths Question 24.
\(\int { \frac { dx }{ { x }^{ 2 }+2x+2 } equals } \)
(a) xtan^-1(x+1)+c
(b) (x+1)tan^-1x+c
(c) tan^-1(x+1)+c
(d) tan^-1x+c
Solution.
(b) \(let\quad I=\int { \frac { dx }{ { x }^{ 2 }+2x+2 } } =\int { \frac { dx }{ (x+1)^{ 2 }+1 } } \)
= (x+1)tan^-1x+c

Ex 7.4 Class 12 Maths Question 25.
\(\int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } equals } \)
(a) \(\frac { 1 }{ 9 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c\)
(b) \(\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c\)
(c) \(\frac { 1 }{ 3 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c\)
(d) \({ sin }^{ -1 }\left( \frac { 9x-8 }{ 9 } \right) +c\)
Solution.
(b) \(\int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } } =\frac { 1 }{ 2 } \left[ \frac { dx }{ \sqrt { \left( \frac { 9 }{ 8 } \right) ^{ 2 }-\left[ { x }^{ 2 }-{ \frac { 9 }{ 4 } }x+\left( \frac { 9 }{ 8 } \right) ^{ 2 } \right] } } \right] \)
\(\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c\)

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.5

Integrate the rational function in exercises 1 to 21

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Ex 7.5 Class 12 Maths Question 1.
\(\frac { x }{ (x+1)(x+2) }\)
Solution.
let \(\frac { x }{ (x+1)(x+2) }\) ≡ \(\frac { A }{ x+1 } +\frac { B }{ x+2 } \)
⇒ x ≡ A(x+2)+B(x+1)….(i)
putting x = -1 & x = -2 in (i)
we get A = 1,B = 2
\(\therefore \int { \frac { 1 }{ (x+1)(x+2) } dx } =\int { \frac { -1 }{ x+1 } dx } +\int { \frac { 2 }{ x+2 } dx } \)
=-log|x+1| + 2log|x+2|+c

Ex 7.5 Class 12 Maths Question 2.
\(\frac { 1 }{ { x }^{ 2 }-9 } \)
Solution.
let \(\frac { 1 }{ { x }^{ 2 }-9 } =\frac { 1 }{ (x-3)(x+3) } \equiv \frac { A }{ x-3 } +\frac { B }{ x+3 } \)
⇒ x ≡ A(x+3)+B(x-3)…(i)
put x = 3, -3 in (i)
we get \(A=\frac { 1 }{ 6 }\) & \(B=-\frac { 1 }{ 6 }\)
\(\therefore \int { \frac { 1 }{ { x }^{ 2 }-9 } dx } =\frac { 1 }{ 6 } \int { \left[ \frac { 1 }{ x-3 } -\frac { 1 }{ x+3 } \right] dx } \)
\(=\frac { 1 }{ 6 } log\left| \frac { x-3 }{ x+3 } \right| +c\)

Ex 7.5 Class 12 Maths Question 3.
\(\frac { 3x-1 }{ (x-1)(x-2)(x-3) }\)
Solution.
Let \(\frac { 3x-1 }{ (x-1)(x-2)(x-3) } =\frac { A }{ x-1 } +\frac { B }{ x-2 } +\frac { C }{ x-3 } \)
⇒ 3x-1 = A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(-2)…..(i)
put x = 1,2,3 in (i)
we get A = 1,B = -5 & C = 4
\(\therefore I=\int { \frac { 1 }{ x-1 } dx } -5\int { \frac { 1 }{ x-2 } dx } +4\int { \frac { 1 }{ x-3 } dx } \)
=log|x-1| – 5log|x-2| + 4log|x+3| + C

Ex 7.5 Class 12 Maths Question 4.
\(\frac { x }{ (x-1)(x-2)(x-3) }\)
Solution.
let \(\frac { x }{ (x-1)(x-2)(x-3) } =\frac { A }{ x-1 } +\frac { B }{ x-2 } +\frac { C }{ x-3 } \)
⇒ x ≡ A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)…(i)
put x = 1,2,3 in (i)
\(A=\frac { 1 }{ 2 } ,B=-2,C=\frac { 3 }{ 2 } \)
\(\therefore I=\frac { 1 }{ 2 } \int { \frac { dx }{ x-1 } } -2\int { \frac { dx }{ x-2 } } +\frac { 3 }{ 2 } \int { \frac { dx }{ x-3 } } \)
\(=\frac { 1 }{ 2 } log|x-1|-2log|x-2|+\frac { 3 }{ 2 } log|x-3|+c \)

Ex 7.5 Class 12 Maths Question 5.
\(\frac { 2x }{ { x }^{ 2 }+3x+2 } \)
Solution.
let \(\frac { 2x }{ { x }^{ 2 }+3x+2 } =\frac { 2x }{ (x+1)(x+2) } =\frac { A }{ x+1 } +\frac { B }{ x+2 } \)
⇒ 2x = A(x+2)+B(x+1)…(i)
put x = -1, -2 in (i)
we get A = -2, B = 4
\(\therefore \int { \frac { 2x }{ { x }^{ 2 }+3x+2 } dx } =-2\int { \frac { dx }{ x+1 } } +4\int { \frac { dx }{ x+2 } } \)
=-2log|x+1|+4log|x+2|+c

Ex 7.5 Class 12 Maths Question 6.
\(\frac { 1-{ x }^{ 2 } }{ x(1-2x) } \)
Solution.
\(\frac { 1-{ x }^{ 2 } }{ (x-2{ x }^{ 2 }) } \) is an improper fraction therefore we
convert it into a proper fraction. Divide 1 – x² by x – 2x² by long division.

Ex 7.5 Class 12 Maths Question 7.
\(\frac { x }{ \left( { x }^{ 2 }+1 \right) \left( x-1 \right) } \)
Solution.
let \(\frac { x }{ \left( { x }^{ 2 }+1 \right) \left( x-1 \right) } =\frac { A }{ x-1 } +\frac { Bx+C }{ { x }^{ 2 }+1 } \)
⇒ x = A(x²+1)+(Bx+C)(x-1)
Put x = 1,0
⇒ \(A=\frac { 1 }{ 2 } C=\frac { 1 }{ 2 } \Rightarrow B=-\frac { 1 }{ 2 } \)
\(\therefore I=\frac { 1 }{ 2 } \int { \frac { dx }{ x-1 } } -\frac { 1 }{ 2 } \int { \frac { x }{ { x }^{ 2 }+1 } dx } +\frac { 1 }{ 2 } \int { \frac { dx }{ { x }^{ 2 }+1 } } \)
\(=\frac { 1 }{ 2 } log(x-1)-\frac { 1 }{ 4 } log({ x }^{ 2 }+1)+\frac { 1 }{ 2 } { tan }^{ -1 }x+c \)

Ex 7.5 Class 12 Maths Question 8.
\(\frac { x }{ { \left( x-1 \right) }^{ 2 }\left( x+2 \right) } \)
Solution.
\(\frac { x }{ { \left( x-1 \right) }^{ 2 }\left( x+2 \right) } =\frac { A }{ x-1 } +\frac { B }{ { \left( x-1 \right) }^{ 2 } } +\frac { C }{ x+2 } \)
⇒ x ≡ A(x-1)(x+2)+B(x+2)+C(x-1)² …(i)
put x = 1, -2
we get \(B=\frac { 1 }{ 3 } ,C=\frac { -2 }{ 9 } \)
\(\therefore I=\frac { 2 }{ 9 } \int { \frac { 1 }{ x-1 } dx } +\frac { 1 }{ 3 } \int { \frac { 1 }{ { (x-1) }^{ 2 } } dx } -\frac { 2 }{ 9 } \int { \frac { 1 }{ x+2 } dx } \)
\(=\frac { 2 }{ 9 } log\left| \frac { x-1 }{ x+2 } \right| -\frac { 1 }{ 3\left( x-1 \right) } +c\)

Ex 7.5 Class 12 Maths Question 9.
\(\frac { 3x+5 }{ { x }^{ 3 }-{ x }^{ 2 }-x+1 } \)
Solution.
let \(\frac { 3x+5 }{ { x }^{ 2 }(x-1)-1(x-1) } \)
\(\frac { 3x+5 }{ (x-1)^{ 2 }(x+1) } =\frac { A }{ x-1 } +\frac { B }{ { (x-1) }^{ 2 } } +\frac { C }{ x+1 } \)
⇒ 3x+5 = A(x-1)(x+1)+B(x+1)+C(x-1)
put x = 1,-1,0
we get \(B=4,C=\frac { 1 }{ 2 } ,A=-\frac { 1 }{ 2 } \)
\(\therefore I=-\frac { 1 }{ 2 } \int { \frac { dx }{ (x-1) } } +4\frac { dx }{ { (x-1) }^{ 2 } } +\frac { 1 }{ 2 } \int { \frac { dx }{ x+1 } } \)
\(=\frac { 1 }{ 2 } log\left| \frac { x+1 }{ x-1 } \right| -\frac { 4 }{ x-1 } +c\)

Ex 7.5 Class 12 Maths Question 10.
\(\frac { 2x-3 }{ ({ x }^{ 2 }-1)(2x+3) } \)
Solution.
\(\frac { 2x-3 }{ ({ x }^{ 2 }-1)(2x+3) } =\frac { 2x-3 }{ (x-1)(x+1)(2x+3) } \)

Ex 7.5 Class 12 Maths Question 11.
\(\frac { 5x }{ (x-1)({ x }^{ 2 }-4) } \)
Solution.
let \(\frac { 5x }{ (x-1)({ x }^{ 2 }-4) } =\frac { 5x }{ (x+1)(x+2)(x-2) } \)

Ex 7.5 Class 12 Maths Question 12.
\(\frac { { x }^{ 3 }+x+1 }{ { x }^{ 2 }-1 } \)
Solution.
\(\frac { { x }^{ 3 }+x+1 }{ { x }^{ 2 }-1 } =x+\frac { 2x+1 }{ (x+1)(x-1) } \)

Ex 7.5 Class 12 Maths Question 13.
\(\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) } \)
Solution.
\(\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) } =\frac { A }{ 1-x } +\frac { Bx+C }{ 1+{ x }^{ 2 } } \)
⇒ 2 = A(1+x²) + (Bx+C)(1 -x) …(i)
Putting x = 1 in (i), we get; A = 1
Also 0 = A – B and 2 = A + C ⇒B = A = 1 & C = 1

Ex 7.5 Class 12 Maths Question 14.
\(\frac { 3x-1 }{ { (x+2) }^{ 2 } } \)
Solution.
\(\frac { 3x-1 }{ { (x+2) }^{ 2 } } \equiv \frac { A }{ x+1 } +\frac { B }{ { (x+2) }^{ 2 } } \)
=>3x – 1 = A(x + 2) + B …(i)
Comparing coefficients A = -1 and B = -7
\(\therefore \int { \frac { 3x-1 }{ { (x+2) }^{ 2 } } dx } =3\int { \frac { dx }{ x+2 } } -7\int { \frac { dx }{ { (x+2) }^{ 2 } } } \)
\(=3log|x+2|+\frac { 7 }{ x+2 } +c\)

Ex 7.5 Class 12 Maths Question 15.
\(\frac { 1 }{ { x }^{ 4 }-1 } \)
Solution.
\(\frac { 1 }{ { x }^{ 4 }-1 } =\frac { A }{ x+1 } +\frac { B }{ x-1 } +\frac { Cx+D }{ { x }^{ 2 }+1 } \)
⇒ 1 ≡ A(x-1)(x²+1) + B(x+1)(x²+1) + (Cx+D)(x+1)(x-1) ….(i)

Ex 7.5 Class 12 Maths Question 16.
\(\frac { 1 }{ x({ x }^{ n }+1) } \)
[Hint : multiply numerator and denominator by x^n-1 and put xⁿ = t ]
Solution.
\(\frac { { x }^{ n-1 } }{ x.{ x }^{ n-1 }({ x }^{ n }+1) } =\frac { { x }^{ n-1 } }{ { x }^{ n }({ x }^{ n }+1) } \)

Ex 7.5 Class 12 Maths Question 17.
\(\frac { cosx }{ (1-sinx)(2-sinx) } \)
Solution.
put sinx = t
so that cosx dx = dt
\(\therefore I=\int { \frac { 1 }{ (1-t)(2-t) } dt } \)

Ex 7.5 Class 12 Maths Question 18.
\(\frac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) } \)
Solution.
put x²=y
\(I=1-\frac { 2(2y+5) }{ (y+3)(y+4) } \)

Ex 7.5 Class 12 Maths Question 19.
\(\frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) } \)
Solution.
put x²=y
so that 2xdx = dy
\(\therefore \int { \frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) } dx } =\int { \frac { dy }{ (y+1)(y+3) } } \)

Ex 7.5 Class 12 Maths Question 20.
\(\frac { 1 }{ x({ x }^{ 4 }-1) } \)
Solution.
put x⁴ = t
so that 4x³ dx = dt

Ex 7.5 Class 12 Maths Question 21.
\(\frac { 1 }{ { e }^{ x }-1 } \)
Solution.
Let e^x = t ⇒ e^x dx = dt
⇒ \(dx=\frac { dt }{ t }\)

Ex 7.5 Class 12 Maths Question 22.
choose the correct answer in each of the following :
\(\int { \frac { xdx }{ (x-1)(x-2) } equals } \)
(a) \(log\left| \frac { { (x-1) }^{ 2 } }{ x-2 } \right| +c\)
(b) \(log\left| \frac { { (x-2) }^{ 2 } }{ x-1 } \right| +c\)
(c) \(log\left| \left( \frac { x-{ 1 }^{ 2 } }{ x-2 } \right) \right| +c\)
(d) log|(x-1)(x-2)|+c
Solution.
(b) \(\int { \frac { x }{ (x-1)(x-2) } dx } =\int { \left[ \frac { -1 }{ x-1 } +\frac { 2 }{ x-2 } \right] dx } \)
\(log\left| \frac { { (x-2) }^{ 2 } }{ x-1 } \right| +c\)

Ex 7.5 Class 12 Maths Question 23.
\(\int { \frac { dx }{ x({ x }^{ 2 }+1) } equals } \)
(a) \(log|x|-\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c \)
(b) \(log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c \)
(c) \(-log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c\)
(d) \(\frac { 1 }{ 2 } log|x|+log({ x }^{ 2 }+1)+c \)
Solution.
(a) let \(\frac { 1 }{ x\left( { x }^{ 2 }+1 \right) } =\frac { A }{ x } +\frac { Bx+C }{ { x }^{ 2 }+1 } \)
⇒ 1 = A(x²+1)+(Bx+C)(x)

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Integrate the functions in Exercises 1 to 22.

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Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution.
By part integration
∫x sinx dx = x(-cosx) – ∫1(-cosx)dx
=-x cosx + ∫cosxdx
=-x cosx + sinx + c

Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution.
∫x sin3x dx = \(x\left( -\frac { cos3x }{ 3 } \right) -\int { 1 } .\left( \frac { -cos3x }{ 3 } \right) dx\)
\(=-\frac { 1 }{ 3 } x\quad cos3x+\frac { 1 }{ 9 } sin3x+c\)

Ex 7.6 Class 12 Maths Question 3.
\({ x }^{ 2 }{ e }^{ x }\)
Solution.
\(\int { { x }^{ 2 }{ e }^{ x } } dx={ x }^{ 2 }{ e }^{ x }-2{ x }{ e }^{ x }+2{ e }^{ x }+c\)
\(={ e }^{ x }\left( { x }^{ 2 }-2x+2 \right) +c\)

Ex 7.6 Class 12 Maths Question 4.
x logx
Solution.
\(\int { xlogx\quad dx } =logx\int { xdx } -\int { \left[ \frac { d }{ dx } (logx)\int { xdx } \right] dx } \)
\(=\frac { { x }^{ 2 } }{ 2 } logx-\frac { 1 }{ 2 } \int { x\quad dx } =\frac { { x }^{ 2 } }{ 2 } logx-\frac { 1 }{ 4 } { x }^{ 2 }+c \)

Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution.
\(\int { x\quad log2xdx } =(log2x)\frac { { x }^{ 2 } }{ 2 } -\int { \frac { 1 }{ 2x } } .2\left( \frac { { x }^{ 2 } }{ 2 } \right) dx\)
\(=\frac { { x }^{ 2 } }{ 2 } log|2x|-\frac { 1 }{ 2 } \int { xdx } =\frac { { x }^{ 2 } }{ 2 } log|2x|-\frac { { x }^{ 2 } }{ 4 } +c\)

Ex 7.6 Class 12 Maths Question 6.
\({ x }^{ 2 }logx\)
Solution.
\(\int { { x }^{ 2 }logxdx } =log|x|\left( \frac { { x }^{ 3 } }{ 3 } \right) -\int { \frac { 1 }{ x } } \left( \frac { { x }^{ 3 } }{ 3 } \right) dx\)
\(=\frac { { x }^{ 3 } }{ 3 } log|x|-\frac { 1 }{ 3 } \int { { x }^{ 2 }dx } =\frac { { x }^{ 3 } }{ 3 } log|x|-\frac { { x }^{ 3 } }{ 9 } +c\)

Ex 7.6 Class 12 Maths Question 7.
\(x\quad { sin }^{ -1 }x\)
Solution.
\(I=x\quad { sin }^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx\)

Ex 7.6 Class 12 Maths Question 8.
\(x\quad { tan }^{ -1 }x\)
Solution.
\(I=x\quad { tan}^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1+{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx\)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \left( 1-\frac { 1 }{ 1+{ x }^{ 2 } } \right) dx } \)
\(=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } x+\frac { 1 }{ 2 } { tan }^{ -1 }x+c\)

Ex 7.6 Class 12 Maths Question 9.
\(x\quad { cos }^{ -1 }x\)
Solution.
let I = \(\int { x } { cos }^{ -1 }xdx=\int { { cos }^{ -1 }x } .xdx\)

Ex 7.6 Class 12 Maths Question 10.
\({ (sin }^{ -1 }{ x })^{ 2 }\)
Solution.
\(put\quad { sin }^{ -1 }x=\theta \Rightarrow x=sin\theta \Rightarrow dx=cos\theta d\theta \)

Ex 7.6 Class 12 Maths Question 11.
\(\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } \)
Solution.
\(put\quad { cos }^{ -1 }x=t\quad so\quad that\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx=dt\)

Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution.
∫x sec²x dx =x(tanx)-∫1.tanx dx
= x tanx+log cosx+c

Ex 7.6 Class 12 Maths Question 13.
\({ ta }n^{ -1 }x\)
Solution.
\(\int { { tan }^{ -1 }xdx } =x{ tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \frac { 2x }{ 1+{ x }^{ 2 } } dx } \)
\(=x{ tan }^{ -1 }x-\frac { 1 }{ 2 } log|1+{ x }^{ 2 }|+c \)

Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution.
∫x(logx)² dx
\(=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\left[ (logx)\frac { { x }^{ 2 } }{ 2 } -\int { \frac { 1 }{ x } \frac { { x }^{ 2 } }{ 2 } dx } \right] \)
\(=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\frac { { x }^{ 2 } }{ 2 } logx+\frac { 1 }{ 4 } { x }^{ 2 }+c\)

Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution.
∫(x²+1)logx dx
\(=logx\left( \frac { { x }^{ 3 } }{ 3 } +x \right) -\int { \frac { 1 }{ x } \left( \frac { { x }^{ 3 } }{ 3 } +x \right) dx } \)
\(=\left( \frac { { x }^{ 3 } }{ 3 } +x \right) logx-\frac { { x }^{ 3 } }{ 9 } -x+c\)

Ex 7.6 Class 12 Maths Question 16.
\({ e }^{ x }(sinx+cosx)\)
Solution.
\(put\quad { e }^{ x }sinx=t\Rightarrow { e }^{ x }(sinx+cosx)dx=dt\)
\(\therefore \int { { e }^{ x }(sinx+cosx)dx } =\int { dt } =t+c\)
\(={ e }^{ x }sinx+c\)

Ex 7.6 Class 12 Maths Question 17.
\(\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } \)
Solution.
\(\int { \frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } } \)

Ex 7.6 Class 12 Maths Question 18.
\(\frac { { e }^{ x }(1+sinx) }{ 1+cosx } \)
Solution.
\(I=\int { { e }^{ x } } \left[ \frac { 1+2sin\frac { x }{ 2 } cos\frac { x }{ 2 } }{ 2{ cos }^{ 2 }\frac { x }{ 2 } } \right] dx\)

Ex 7.6 Class 12 Maths Question 19.
\({ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) \)
Solution.
put \(\frac { { e }^{ x } }{ x } =t\Rightarrow { e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) dx=dt\)
\(\therefore I=\int { dt } =t+c=\frac { { e }^{ x } }{ x } +c\)

Ex 7.6 Class 12 Maths Question 20.
\(\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } } \)
Solution.
\(I=\int { { e }^{ x }\left[ \frac { 1 }{ { (x-1) }^{ 2 } } -\frac { 2 }{ { (x-1) }^{ 3 } } \right] dx } \)

Ex 7.6 Class 12 Maths Question 21.
\({ e }^{ 2x }sinx\)
Solution.
let \(I=\int { { e }^{ 2x }sinx } \)
\(={ e }^{ 2x }(-cosx)-\int { 2{ e }^{ 2x }(-cosx)dx } \)

Ex 7.6 Class 12 Maths Question 22.
\({ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) \)
Solution.
Put x = tan t
so that dx = sec² t dt

Choose the correct answer in exercise 23 and 24

Ex 7.6 Class 12 Maths Question 23.
\(\int { { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx\quad equals\)
(a) \(\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c\)
(b) \(\frac { 1 }{ 3 } +{ e }^{ { x }^{ 2 } }+c\)
(c) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+c\)
(d) \(\frac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+c\)
Solution.
(a) let x³ = t
⇒3x² dx = dt
\(\therefore \int { { x }^{ 2 }{ e }^{ { x }^{ 3 } }dx } =\frac { 1 }{ 3 } \int { { e }^{ t }dt } =\frac { 1 }{ 3 } { e }^{ t }+c=\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c\)

Ex 7.6 Class 12 Maths Question 24.
\(\int { { e }^{ x }secx(1+tanx) } dx\quad equals\)
(a) \({ e }^{ x }cosx+c\)
(b) \({ e }^{ x }secx+c\)
(c) \({ e }^{ x }sinx+c\)
(d) \({ e }^{ x }tanx+c\)
Solution.
(b) \(\int { { e }^{ x }(secx+secx\quad tanx)dx } ={ e }^{ x }secx+c\)

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.7

Integral the function in exercises 1 to 9

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Ex 7.7 Class 12 Maths Question 1.
\(\sqrt { 4-{ x }^{ 2 } } \)
Solution.
\(let\quad I=\int { \sqrt { 4-{ x }^{ 2 } } } dx=\int { \sqrt { { (2) }^{ 2 }-{ x }^{ 2 } } dx } \)
\(=\frac { x\sqrt { 4-{ x }^{ 2 } } }{ 2 } +2{ sin }^{ -1 }\left( \frac { x }{ 2 } \right) +c \)

Ex 7.7 Class 12 Maths Question 2.
\(\sqrt { 1-{ 4x }^{ 2 } } \)
Solution.
\(\int { \sqrt { 1-{ 4x }^{ 2 } } } dx=2\int { \sqrt { { \left( \frac { 1 }{ 2 } \right) }^{ 2 }-{ x }^{ 2 } } } dx\)
\(=\frac { x\sqrt { 1-{ 4x }^{ 2 } } }{ 2 } +\frac { 1 }{ 4 } { sin }^{ -1 }(2x)+c\)

Ex 7.7 Class 12 Maths Question 3.
\(\sqrt { { x }^{ 2 }+4x+6 } \)
Solution.
\(\int { \sqrt { { x }^{ 2 }+4x+6 } } dx=\int { \sqrt { { (x+2) }^{ 2 }+{ (\sqrt { 2 } ) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x+6 } +log\left| (x+2)+\sqrt { { x }^{ 2 }+4x+6 } \right| +c\)

Ex 7.7 Class 12 Maths Question 4.
\(\sqrt { { x }^{ 2 }+4x+1 } \)
Solution.
\(\int { \sqrt { { x }^{ 2 }+4x+1 } } dx=\int { \sqrt { { (x+2) }^{ 2 }-{ (\sqrt { 3 } ) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x+1 } -\frac { 3 }{ 2 } log\left| (x+2)+\sqrt { { x }^{ 2 }+4x+1 } \right| +c\)

Ex 7.7 Class 12 Maths Question 5.
\(\sqrt { 1-4x-{ x }^{ 2 } } \)
Solution.
\(\int { \sqrt { 1-4x-{ x }^{ 2 } } } dx=\int { \sqrt { { (5) }^{ 2 }-{ (x+2) }^{ 2 } } dx } \)
\(=\frac { x+2 }{ 2 } \sqrt { 5-{ (x+2) }^{ 2 } } dx \)

Ex 7.7 Class 12 Maths Question 6.
\(\sqrt { { x }^{ 2 }+4x-5 } \)
Solution.
\(\int { \sqrt { { x }^{ 2 }+4x-5 } } dx=\int { \sqrt { { (x+2) }^{ 2 }-{ (3) }^{ 2 } } } dx\)
\(=\frac { x+2 }{ 2 } \sqrt { { x }^{ 2 }+4x-5 } -\frac { 9 }{ 2 } log|x+2+\sqrt { { x }^{ 2 }+4x-5 } |+c\)

Ex 7.7 Class 12 Maths Question 7.
\(\sqrt { 1+3x-{ x }^{ 2 } } \)
Solution.
\(\int { \sqrt { 1-\left( { x }^{ 2 }-3x \right) } } dx\)
\(=\int { \sqrt { { \left( \frac { \sqrt { 13 } }{ 2 } \right) }^{ 2 }-{ \left( x-\frac { 3 }{ 2 } \right) }^{ 2 } } } dx\)
\(=\frac { 2x-3 }{ 4 } \sqrt { 1+3x-{ x }^{ 2 } } +\frac { 13 }{ 8 } { sin }^{ -1 }\left[ \frac { 2x-3 }{ \sqrt { 3 } } \right] +c\)

Ex 7.7 Class 12 Maths Question 8.
\(\sqrt { { x }^{ 2 }+3x } \)
Solution.
\(\int { \sqrt { { x }^{ 2 }+3x } } dx=\int { \sqrt { { \left( x+\frac { 3 }{ 2 } \right) }^{ 2 }-{ \left( \frac { 3 }{ 2 } \right) }^{ 2 } } } dx\)
\(=\frac { 2x+3 }{ 4 } \sqrt { { x }^{ 2 }+3x } -\frac { 9 }{ 8 } log\left| x+\frac { 3 }{ 2 } +\sqrt { { x }^{ 2 }+3x } \right| +c\)

Ex 7.7 Class 12 Maths Question 9.
\(\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } \)
Solution.
\(\int { \sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } } dx=\frac { 1 }{ 3 } \int { \sqrt { { x }^{ 2 }+{ 3 }^{ 2 } } } \)
\(=\frac { 1 }{ 6 } \left[ x\sqrt { { x }^{ 2 }+9 } +9log|x+\sqrt { { x }^{ 2 }+9 } | \right] +c\)

Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.
\(\int { \sqrt { 1+{ x }^{ 2 } } } dx\quad is\quad equal\quad to\)
(a) \(\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+c\)
(b) \(\frac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c\)
(c) \(\frac { 2 }{ 3 } x{ \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c\)
(d) \(\frac { { x }^{ 2 } }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } { x }^{ 2 }log\left| x+\sqrt { 1+{ x }^{ 2 } } \right| +c\)
Solution.
(a) \(\int { \sqrt { 1+{ x }^{ 2 } } } dx\)
\(=\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log(x+\sqrt { 1+{ x }^{ 2 } } )+c\)

Ex 7.7 Class 12 Maths Question 11.
\(\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to\)

Solution.
(d) \(\int { \sqrt { { x }^{ 2 }-8x+7 } } dx=\int { \sqrt { { (x-4) }^{ 2 }-{ (3) }^{ 2 } } } dx\)
\(=\frac { x-4 }{ 2 } \sqrt { { x }^{ 2 }-8x+7 } -\frac { 9 }{ 2 } log\left| (x-4)+\sqrt { { x }^{ 2 }+8x+7 } \right| +c\)

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.8

Evaluate the following definite integral as limit of sums.

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Ex 7.8 Class 12 Maths Question 1.
\(\int _{ a }^{ b }{ x\quad dx } \)
Solution.
on comparing
\(\int _{ a }^{ b }{ x\quad dx } \quad with\quad \int _{ a }^{ b }{ f(x)dx } \)
we have

Ex 7.8 Class 12 Maths Question 2.
\(\int _{ 0 }^{ 5 }{ (x+1)dx } \)
Solution.
on comparing
\(\int _{ 0 }^{ 5 }{ (x+1)dx } \quad with\quad \int _{ 0 }^{ 5 }{ f(x)dx } \)
we have f(x) = x+1, a = 0, b = 5
and nh = b-a = 5-0 = 5

Ex 7.8 Class 12 Maths Question 3.
\(\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx\)
Solution.
compare
\(\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx\)
we have

Ex 7.8 Class 12 Maths Question 4.
\(\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx\)
Solution.
compare
\(\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx\)
we have f(x) = x²-x and a = 1, b = 4

Ex 7.8 Class 12 Maths Question 5.
\(\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad \)
Solution.
compare
\(\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad with\quad \int _{ a }^{ b }{ f({ x }) } dx\)
we have

Ex 7.8 Class 12 Maths Question 6.
\(\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad \)
Solution.
let f(x) = x + e^2x,
a = 0, b = 4
and nh = b – a = 4 – 0 = 4

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.9

Evaluate the definite integrals in Exercise 1 to 20.

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Ex 7.9 Class 12 Maths Question 1.
\(\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad \)
Solution.
\({ =\left[ \frac { { x }^{ 2 } }{ 2 } +x \right] }_{ -1 }^{ 1 }=\frac { 1 }{ 2 } (1-1)+(1+1)\quad =2\)

Ex 7.9 Class 12 Maths Question 2.
\(\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx } \)
Solution.
\(={ \left[ log\quad x \right] }_{ 2 }^{ 3 }\quad =log3-log2\quad =log\frac { 3 }{ 2 } \)

Ex 7.9 Class 12 Maths Question 3.
\(\int _{ 1 }^{ 2 }{ \left( { 4x }^{ 3 }-{ 5x }^{ 2 }+6x+9 \right) dx } \)
Solution.
\(={ \left[ \frac { { 4x }^{ 4 } }{ 4 } -\frac { { 5x }^{ 3 } }{ 3 } +\frac { { 6x }^{ 2 } }{ 2 } +9x \right] }_{ 1 }^{ 2 }\)
\(={ \left[ { x }^{ 4 }-\frac { 5 }{ 3 } { x }^{ 3 }+{ 3x }^{ 2 }+9x \right] }_{ 1 }^{ 2 }\quad =\frac { 64 }{ 3 } \)

Ex 7.9 Class 12 Maths Question 4.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx } \)
Solution.
\(={ \left[ -\frac { 1 }{ 2 } cos2x \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\quad =\frac { 1 }{ 2 } \)

Ex 7.9 Class 12 Maths Question 5.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx } \)
Solution.
\(={ \left[ \frac { 1 }{ 2 } sin2x \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\quad =0\)

Ex 7.9 Class 12 Maths Question 6.
\(\int _{ 4 }^{ 5 }{ { e }^{ x }dx } \)
Solution.
\(={ \left[ { e }^{ x } \right] }_{ 4 }^{ 5 }\quad ={ e }^{ 5 }-{ e }^{ 4 }\)

Ex 7.9 Class 12 Maths Question 7.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx } \)
Solution.
\(={ \left[ log\quad secx \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\quad =\frac { 1 }{ 2 } log2\)

Ex 7.9 Class 12 Maths Question 8.
\(\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx } \)
Solution.
\(=log{ \left( cosecx-cotx \right) }_{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }\)
\(=log(\sqrt { 2 } -1)-log(2-\sqrt { 3 } )\quad =log\left( \frac { \sqrt { 2 } -1 }{ 2-\sqrt { 3 } } \right) \)

Ex 7.9 Class 12 Maths Question 9.
\(\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } } \)
Solution.
\(={ sin }^{ -1 }(1)-{ sin }^{ -1 }(0)\quad =\frac { \pi }{ 2 } \)

Ex 7.9 Class 12 Maths Question 10.
\(\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } } \)
Solution.
\(={ \left[ { tan }^{ -1 }x \right] }_{ 0 }^{ 1 }\quad ={ tan }^{ -1 }(1)-{ ta }n^{ -1 }(0)\quad =\frac { \pi }{ 4 } \)

Ex 7.9 Class 12 Maths Question 11.
\(\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } } \)
Solution.
\(={ \left[ \frac { 1 }{ 2 } log\left( \frac { x-1 }{ x+1 } \right) \right] }_{ 2 }^{ 3 }\quad =\frac { 1 }{ 2 } log\frac { 3 }{ 2 } \)

Ex 7.9 Class 12 Maths Question 12.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx\)
Solution.
\(=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { \frac { 1+cos2x }{ 2 } } } dx=\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }=\frac { \pi }{ 4 } \)

Ex 7.9 Class 12 Maths Question 13.
\(\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx\)
Solution.
\(=\frac { 1 }{ 2 } \int _{ 2 }^{ 3 }{ \frac { 2x }{ { x }^{ 2 }+1 } } dx\quad =\frac { 1 }{ 2 } { \left[ log\left( { x }^{ 2 }+1 \right) \right] }_{ 2 }^{ 3 }\quad =\frac { 1 }{ 2 } log2\)

Ex 7.9 Class 12 Maths Question 14.
\(\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx } \)
Solution.
\(=\frac { 1 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { 10x }{ { 5x }^{ 2 }+1 } dx } +\frac { 3 }{ 5 } \int _{ 0 }^{ 1 }{ \frac { dx }{ { { x }^{ 2 }+\left[ \frac { 1 }{ \sqrt { 5 } } \right] }^{ 2 } } } \)

Ex 7.9 Class 12 Maths Question 15.
\(\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx } \)
Solution.
let x² = t ⇒ 2xdx = dt
when x = 0, t = 0 & when x = 1,t = 1
\(\therefore I=\frac { 1 }{ 2 } \int _{ 0 }^{ 1 }{ { e }^{ t }dt } \quad =\frac { 1 }{ 2 } { \left( { e }^{ t } \right) }_{ 0 }^{ 1 }\quad =\frac { 1 }{ 2 } [e-1]\)

Ex 7.9 Class 12 Maths Question 16.
\(\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx } \)
Solution.
\(\int _{ 1 }^{ 2 }{ \left( 5-\frac { 20x+15 }{ { x }^{ 2 }+4x+3 } \right) dx } \)

Ex 7.9 Class 12 Maths Question 17.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx } \)
Solution.
\(={ \left[ 2tanx+\frac { { x }^{ 4 } }{ 4 } +2x \right] }_{ 0 }^{ \frac { \pi }{ 4 } }\)

Ex 7.9 Class 12 Maths Question 18.
\(\int _{ 0 }^{ \pi }{ \left( { sin }^{ 2 }\frac { x }{ 2 } -{ cos }^{ 2 }\frac { x }{ 2 } \right) } dx\)
Solution.
\(=-\int _{ 0 }^{ \pi }{ cosx } dx\quad =-{ \left[ sinx \right] }_{ 0 }^{ \pi }-(0-0)\quad =0\)

Ex 7.9 Class 12 Maths Question 19.
\(\int _{ 0 }^{ 2 }{ \frac { 6x+3 }{ { x }^{ 2 }+4 } } dx\)
Solution.
\(=\int _{ 0 }^{ 2 }{ \frac { 6x }{ { x }^{ 2 }+4 } } dx+\int _{ 0 }^{ 2 }{ \frac { 3 }{ { x }^{ 2 }+4 } dx } \)

Ex 7.9 Class 12 Maths Question 20.
\(\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx } \)
Solution.
\(=\int _{ 0 }^{ 1 }{ { xe }^{ x }dx } +\int _{ 0 }^{ 1 }{ sin\frac { \pi x }{ 4 } } dx\)

Ex 7.9 Class 12 Maths Question 21.
\(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } \quad equals } \)
(a) \(\frac { \pi }{ 3 } \)
(b) \(\frac { 2\pi }{ 3 } \)
(c) \(\frac { \pi }{ 6 } \)
(d) \(\frac { \pi }{ 12 } \)
Solution.
(d) \(\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } } \quad ={ \left[ { tan }^{ -1 }x \right] }_{ 1 }^{ \sqrt { 3 } }\quad =\frac { \pi }{ 3 } -\frac { \pi }{ 4 } \quad =\frac { \pi }{ 12 } \)

Ex 7.9 Class 12 Maths Question 22.
\(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } equals } \)
(a) \(\frac { \pi }{ 6 }\)
(b) \(\frac { \pi }{ 12 }\)
(c) \(\frac { \pi }{ 24 }\)
(d) \(\frac { \pi }{ 4 }\)
Solution.
(c) \(\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } } \quad =\frac { 1 }{ 9 } \int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ { \left( \frac { 2 }{ 3 } \right) }^{ 2 }+{ x }^{ 2 } } } \)
\(=\frac { 1 }{ 6 } { \left[ { tan }^{ -1 }\left( \frac { 3x }{ 2 } \right) \right] }_{ 0 }^{ \frac { 2 }{ 3 } }\quad =\frac { 1 }{ 6 } \times \frac { \pi }{ 4 } \quad =\frac { \pi }{ 24 } \)

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.10

Evaluate the integrals in Exercises 1 to 8 using substitution.

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Ex 7.10 Class 12 Maths Question 1.
\(\int _{ 0 }^{ 1 }{ \frac { x }{ { x }^{ 2 }+1 } } dx=I\)
Solution.
Let x² + 1 = t
⇒2xdx = dt
when x = 0, t = 1 and when x = 1, t = 2
\(\therefore I=\frac { 1 }{ 2 } \int _{ 0 }^{ 1 }{ \frac { dt }{ t } } ={ \left[ \frac { 1 }{ 2logt } \right] }_{ 1 }^{ 2 }\quad =\frac { 1 }{ 2 } log2\)

Ex 7.10 Class 12 Maths Question 2.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sqrt { sin\phi } { cos }^{ 5 }\phi d\phi =I } \)
Solution.
\(I=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \sqrt { sin\phi } { (1-{ sin }^{ 2 }) }^{ 2 }cos\phi d\phi } \)
put sinφ = t,so that cosφdφ = dt

Ex 7.10 Class 12 Maths Question 3.
\(\int _{ 0 }^{ 1 }{ { sin }^{ -1 } } \left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) dx=I\)
Solution.
let x = tanθ =>dx = sec²θ dθ
when x = 0 => θ = 0
and when x = 1 => \(\theta \frac { \pi }{ 4 } \)
\(\frac { 1 }{ 2 }\)

Ex 7.10 Class 12 Maths Question 4.
\(\int _{ 0 }^{ 2 }{ x\sqrt { x+2 } } dx=I(say)(put\quad x+2={ t }^{ 2 })\)
Solution.
let x+2 = t =>dx = dt
when x = 0,t = 2 and when x = 2, t = 4

Ex 7.10 Class 12 Maths Question 5.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx\quad dx }{ 1+{ cos }^{ 2 }x } =I } \)
Solution.
put cosx = t
so that -sinx dx = dt
when x = 0, t = 1; when \(x=\frac { \pi }{ 2 }\), t = 0
\(\therefore I=\int _{ 1 }^{ 0 }{ \frac { -dt }{ 1+{ t }^{ 2 } } =-{ \left[ { tan }^{ -1 }t \right] }_{ 1 }^{ 0 } } =\frac { \pi }{ 4 } \)

Ex 7.10 Class 12 Maths Question 6.
\(\int _{ 0 }^{ 2 }{ \frac { dx }{ x+4-{ x }^{ 2 } } =I } \)
Solution.
\(\int _{ 0 }^{ 2 }{ \frac { dx }{ x+4-{ x }^{ 2 } } =I } \)

Ex 7.10 Class 12 Maths Question 7.
\(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } =I } \)
Solution.
\(I=\int _{ -1 }^{ 1 }{ \frac { dx }{ { (x+1) }^{ 2 }+{ 2 }^{ 2 } } } =\frac { 1 }{ 2 } { \left[ { tan }^{ -1 }\frac { x+1 }{ 2 } \right] }_{ -1 }^{ 1 }\quad =\frac { \pi }{ 8 } \)

Ex 7.10 Class 12 Maths Question 8.
\(\int _{ 1 }^{ 2 }{ \left[ \frac { 1 }{ x } -\frac { 1 }{ { 2x }^{ 2 } } \right] { e }^{ 2x }dx } =I\)
Solution.
let 2x = t ⇒ 2dx = dt
when x = 1, t = 2 and when x = 2, t = 4
\(I=\int _{ 2 }^{ 4 }{ e } ^{ t }\left( \frac { 1 }{ t } -\frac { 1 }{ { t }^{ 2 } } \right) dt\quad ={ e }^{ t }{ \left[ \frac { 1 }{ t } \right] }_{ 2 }^{ 4 }\quad =\frac { e^{ 2 } }{ 2 } \left[ \frac { { e }^{ 2 } }{ 2 } -1 \right] \)

Choose the correct answer in Exercises 9 and 10

Ex 7.10 Class 12 Maths Question 9.
The value of integral \(\int _{ \frac { 1 }{ 3 } }^{ 1 }{ \frac { { { (x-x }^{ 3 }) }^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx } \) is
(a) 6
(b) 0
(c) 3
(d) 4
Solution.
(a) let I = \(\int _{ \frac { 1 }{ 3 } }^{ 1 }{ \frac { { { (x-x }^{ 3 }) }^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx } \quad =\int _{ \frac { 1 }{ 3 } }^{ 1 }{ \frac { { x }^{ \frac { 1 }{ 3 } }(1-{ x }^{ 2 })^{ \frac { 1 }{ 3 } } }{ { x }^{ 4 } } dx } \)

Ex 7.10 Class 12 Maths Question 10.
\(If\quad f(x)=\int _{ 0 }^{ x }{ tsint,\quad then\quad { f }^{ \prime }(x)\quad is } \)
(a) cosx+xsinx
(b) xsinx
(c) xcosx
(d) sinx+xcosx
Solution.
(b) \(f(x)=\int _{ 0 }^{ x }{ tsint\quad dt } \)
\(=t(-cost)-\int { 1{ \left[ (-cost)dt \right] }_{ 0 }^{ x } } \)
=-x cox+sinx

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.11

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

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Ex 7.11 Class 12 Maths Question 1.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 }x\quad dx } =I\)
Solution.
\(I=\frac { 1 }{ 2 } \int _{ 0 }^{ \frac { \pi }{ 2 } }{ (1+cos2x)dx } =\frac { 1 }{ 2 } { \left[ x+\frac { sin2x }{ 2 } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\quad =\frac { \pi }{ 4 } \)

Ex 7.11 Class 12 Maths Question 2.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx } \)
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { \sqrt { sinx } }{ \sqrt { sinx } +\sqrt { cosx } } dx } \)

Ex 7.11 Class 12 Maths Question 3.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx } \)
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { sin }^{ \frac { 3 }{ 2 } }xdx }{ { sin }^{ \frac { 3 }{ 2 } }x+{ cos }^{ \frac { 3 }{ 2 } }dx } dx } \)

Ex 7.11 Class 12 Maths Question 4.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } } \)
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { cos }^{ 5 }xdx }{ { sin }^{ 5 }x+{ cos }^{ 5 }x } } \)

Ex 7.11 Class 12 Maths Question 5.
\(\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx=I } \)
Solution.
\(I=\int _{ -5 }^{ 5 }{ \left| x+2 \right| dx+\int _{ -2 }^{ 5 }{ \left| x+2 \right| dx } } \)
at x = – 5, x + 2 < 0; at x = – 2, x + 2 = 0; at x = 5, x + 2>0;x + 2<0, x + 2 = 0, x + 2>0

Ex 7.11 Class 12 Maths Question 6.
\(\int _{ 2 }^{ 8 }{ |x-5|dx } =I\)
Solution.
\(\int _{ 2 }^{ 8 }{ |x-5|dx } =I\)

Ex 7.11 Class 12 Maths Question 7.
\(\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I\)
Solution.
\(\int _{ 0 }^{ 1 }{ x(1-x)^{ n }dx } =I\)

Ex 7.11 Class 12 Maths Question 8.
\(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx } \)
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ log(1+tanx)dx } \)

Ex 7.11 Class 12 Maths Question 9.
\(\int _{ 0 }^{ 2 }{ x\sqrt { 2-x } dx=I } \)
Solution.
let 2-x = t
⇒ – dx = dt
when x = 0, t = 2 and when x = 2,t = 0
\(\frac { 1 }{ 2 }\)

Ex 7.11 Class 12 Maths Question 10.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I } \)
Solution.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left( 2logsinx-logsin2x \right) dx=I } \)

Ex 7.11 Class 12 Maths Question 11.
\(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx\)
Solution.
Let f(x) = sin² x
f(-x) = sin² x = f(x)
∴ f(x) is an even function
\(\therefore \int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 } } xdx\quad =2\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \left[ \frac { 1-cos2x }{ 2 } \right] dx } \)
\(={ \left[ x-\frac { sin2x }{ x } \right] }_{ 0 }^{ \frac { \pi }{ 2 } }\therefore I=\frac { \pi }{ 2 } \)

Ex 7.11 Class 12 Maths Question 12.
\(\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } } \)
Solution.
let I = \(\int _{ 0 }^{ \pi }{ \frac { xdx }{ 1+sinx } } \) …(i)

Ex 7.11 Class 12 Maths Question 13.
\(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx\)
Solution.
Let f(x) = sin⁷ xdx
⇒ f(-x) = -sin⁷ x = -f(x)
⇒ f(x) is an odd function of x
⇒ \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ { sin }^{ 7 } } xdx=0\)

Ex 7.11 Class 12 Maths Question 14.
\(\int _{ 0 }^{ 2\pi }{ { cos }^{ 5 } } xdx\)
Solution.
let f(x) = cos⁵ x
⇒ f(2π – x) = cos⁵ x

Ex 7.11 Class 12 Maths Question 15.
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx } \)
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sinx-cosx }{ 1+sinx\quad cosx } dx } \) …(i)

Ex 7.11 Class 12 Maths Question 16.
\(\int _{ 0 }^{ \pi }{ log(1+cosx)dx } \)
Solution.
let I = \(\int _{ 0 }^{ \pi }{ log(1+cosx)dx } \)
then I = \(\int _{ 0 }^{ \pi }{ log[1+cos(\pi -x)]dx } \)

Ex 7.11 Class 12 Maths Question 17.
\(\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx } \)
Solution.
let I = \(\int _{ 0 }^{ a }{ \frac { \sqrt { x } }{ \sqrt { x } +\sqrt { a-x } } dx } \) …(i)

Ex 7.11 Class 12 Maths Question 18.
\(\int _{ 0 }^{ 4 }{ \left| x-1 \right| dx=I } \)
Solution.
\(I=-\int _{ 0 }^{ 1 }{ (x-1)dx } +\int _{ 1 }^{ 4 }{ (x-1)dx } \)
\(=-{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 0 }^{ 1 }+{ \left[ \frac { { x }^{ 2 } }{ 2 } -x \right] }_{ 1 }^{ 4 }=5 \)

Ex 7.11 Class 12 Maths Question 19.
show that \(4\int _{ 0 }^{ a }{ f(x)g(x)dx } =2\int _{ 0 }^{ a }{ f(x)dx } \) if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4
Solution.
let I = \(\int _{ 0 }^{ a }{ f(x)g(x)dx } \)

Ex 7.11 Class 12 Maths Question 20.
The value of \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx } \) is
(a) 0
(b) 2
(c) π
(d) 1
Solution.
(c) let I = \(\int _{ \frac { -\pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ \left( { x }^{ 3 }+xcosx+{ tan }^{ 5 }x+1 \right) dx } \) is

Ex 7.11 Class 12 Maths Question 21.
The value of \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx } \) is
(a) 2
(b) \(\frac { 3 }{ 4 }\)
(c) 0
(d) -2
Solution.
let I = \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ log\left[ \frac { 4+3sinx }{ 4+3sinx } \right] dx } \)

NCERT Class 12 Maths

Class 12 Maths Chapters | Maths Class 12 Chapter 7

Chapterwise NCERT Solutions for Class 12 Maths

NCERT Solutions For Class 12 Maths Chapter 1 Relations and Functions
NCERT Solutions For Class 12 Maths Chapter 2 Inverse Trigonometric Functions
NCERT Solutions For Class 12 Maths Chapter 3 Matrix
NCERT Solutions For Class 12 Maths Chapter 4 Determinants
NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability
NCERT Solutions For Class 12 Maths Chapter 6 Application of Derivatives
NCERT Solutions For Class 12 Maths Chapter 7 Integrals
NCERT Solutions For Class 12 Maths Chapter 8 Application of Integrals
NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations
NCERT Solutions For Class 12 Maths Chapter 10 Vector Algebra
NCERT Solutions For Class 12 Maths Chapter 11 Three-dimensional Geometry
NCERT Solutions For Class 12 Maths Chapter 12 Linear Programming
NCERT Solutions For Class 12 Maths Chapter 13 Probability

NCERT Solutions for Class 12 All Subjects	NCERT Solutions for Class 10 All Subjects
NCERT Solutions for Class 11 All Subjects	NCERT Solutions for Class 9 All Subjects

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