NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 7
Chapter Name	Integrals
Exercise	Ex 7.6
Number of Questions Solved	24
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

Integrate the functions in Exercises 1 to 22.

Question 1.
x sinx
Solution:
By part integration
∫x sinx dx = x(-cosx) – ∫1(-cosx)dx
=-x cosx + ∫cosxdx
=-x cosx + sinx + c

Question 2.
x sin3x
Solution:
∫x sin3x dx = $x\left( -\frac { cos3x }{ 3 } \right) -\int { 1 } .\left( \frac { -cos3x }{ 3 } \right) dx$
$=-\frac { 1 }{ 3 } x\quad cos3x+\frac { 1 }{ 9 } sin3x+c$

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

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

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

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

Question 7.
$x\quad { sin }^{ -1 }x$
Solution:
$I=x\quad { sin }^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 7

Question 8.
$x\quad { tan }^{ -1 }x$
Solution:
$I=x\quad { tan}^{ -1 }x.\left( \frac { { x }^{ 2 } }{ 2 } \right) -\int { \frac { 1 }{ \sqrt { 1+{ x }^{ 2 } } } } .\frac { { x }^{ 2 } }{ 2 } dx$
$=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \left( 1-\frac { 1 }{ 1+{ x }^{ 2 } } \right) dx }$
$=\frac { { x }^{ 2 } }{ 2 } { tan }^{ -1 }x-\frac { 1 }{ 2 } x+\frac { 1 }{ 2 } { tan }^{ -1 }x+c$

Question 9.
$x\quad { cos }^{ -1 }x$
Solution:
let I = $\int { x } { cos }^{ -1 }xdx=\int { { cos }^{ -1 }x } .xdx$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 9

Question 10.
${ (sin }^{ -1 }{ x })^{ 2 }$
Solution:
$put\quad { sin }^{ -1 }x=\theta \Rightarrow x=sin\theta \Rightarrow dx=cos\theta d\theta$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 10

Question 11.
$\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:
$put\quad { cos }^{ -1 }x=t\quad so\quad that\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } dx=dt$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 11

Question 12.
x sec²x
Solution:
∫x sec²x dx =x(tanx)-∫1.tanx dx
= x tanx+log cosx+c

Question 13.
${ ta }n^{ -1 }x$
Solution:
$\int { { tan }^{ -1 }xdx } =x{ tan }^{ -1 }x-\frac { 1 }{ 2 } \int { \frac { 2x }{ 1+{ x }^{ 2 } } dx }$
$=x{ tan }^{ -1 }x-\frac { 1 }{ 2 } log|1+{ x }^{ 2 }|+c$

Question 14.
x(logx)²
Solution:
∫x(logx)² dx
$=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\left[ (logx)\frac { { x }^{ 2 } }{ 2 } -\int { \frac { 1 }{ x } \frac { { x }^{ 2 } }{ 2 } dx } \right]$
$=\frac { { x }^{ 2 } }{ 2 } { (logx) }^{ 2 }-\frac { { x }^{ 2 } }{ 2 } logx+\frac { 1 }{ 4 } { x }^{ 2 }+c$

Question 15.
(x²+1)logx
Solution:
∫(x²+1)logx dx
$=logx\left( \frac { { x }^{ 3 } }{ 3 } +x \right) -\int { \frac { 1 }{ x } \left( \frac { { x }^{ 3 } }{ 3 } +x \right) dx }$
$=\left( \frac { { x }^{ 3 } }{ 3 } +x \right) logx-\frac { { x }^{ 3 } }{ 9 } -x+c$

Question 16.
${ e }^{ x }(sinx+cosx)$
Solution:
$put\quad { e }^{ x }sinx=t\Rightarrow { e }^{ x }(sinx+cosx)dx=dt$
$\therefore \int { { e }^{ x }(sinx+cosx)dx } =\int { dt } =t+c$
$={ e }^{ x }sinx+c$

Question 17.
$\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } }$
Solution:
$\int { \frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 17

Question 18.
$\frac { { e }^{ x }(1+sinx) }{ 1+cosx }$
Solution:
$I=\int { { e }^{ x } } \left[ \frac { 1+2sin\frac { x }{ 2 } cos\frac { x }{ 2 } }{ 2{ cos }^{ 2 }\frac { x }{ 2 } } \right] dx$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 18

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

Question 20.
$\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } }$
Solution:
$I=\int { { e }^{ x }\left[ \frac { 1 }{ { (x-1) }^{ 2 } } -\frac { 2 }{ { (x-1) }^{ 3 } } \right] dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 19

Question 21.
${ e }^{ 2x }sinx$
Solution:
let $I=\int { { e }^{ 2x }sinx }$
$={ e }^{ 2x }(-cosx)-\int { 2{ e }^{ 2x }(-cosx)dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 21

Question 22.
${ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$
Solution:
Put x = tan t
so that dx = sec² t dt
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 22

Choose the correct answer in exercise 23 and 24

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
$\therefore \int { { x }^{ 2 }{ e }^{ { x }^{ 3 } }dx } =\frac { 1 }{ 3 } \int { { e }^{ t }dt } =\frac { 1 }{ 3 } { e }^{ t }+c=\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c$

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) $\int { { e }^{ x }(secx+secx\quad tanx)dx } ={ e }^{ x }secx+c$

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.6

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