NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

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Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 7
Chapter Name	Integrals
Exercise	Ex 7.4
Number of Questions Solved	25
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

Integrate the functions in exercises 1 to 23

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:
$frac { 1 }{ 2 } int { frac { dx }{ sqrt { frac { 1 }{ 4 } +{ x }^{ 2 } } } } =frac { 1 }{ 2 } int { frac { dx }{ sqrt { { left( frac { 1 }{ 2 } right) }^{ 2 }+{ x }^{ 2 } } } }$
$=frac { 1 }{ 2 } logleft| 2x+sqrt { 1+{ 4x }^{ 2 } } right| +c$

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
$int { frac { dx }{ sqrt { { (2-x) }^{ 2 }+1 } } } =-int { frac { dt }{ sqrt { { t }^{ 2 }+1 } } } =-log|t+sqrt { { t }^{ 2 }+1 } |+c$
$=logleft| frac { 1 }{ (2-x)+sqrt { { x }^{ 2 }-4x+5 } } right| +c$

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

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

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

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
$I=frac { 1 }{ 3 } int { frac { dt }{ { t }^{ 2 }+{ { (a }^{ 3 }) }^{ 2 } } =frac { 1 }{ 3 } logleft| t+sqrt { { t }^{ 2 }+{ a }^{ 6 } } right| +c }$
$=frac { 1 }{ 3 } log|{ x }^{ 3 }+sqrt { { a }^{ 6 }+{ x }^{ 6 } } |+c$

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:
$int { frac { 1 }{ sqrt { (x-1)(x-2) } } dx } =int { frac { dx }{ sqrt { { left( x-frac { 3 }{ 2 } right) }^{ 2 }-{ left( frac { 1 }{ 2 } right) }^{ 2 } } } }$
$=logleft| x-frac { 3 }{ 2 } +sqrt { { x }^{ 2 }-3x+2 } right| +c$

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:
$int { frac { dx }{ sqrt { (x-a)(x-b) } } } =int { frac { dx }{ { left( x-frac { a+b }{ 2 } right) }^{ 2 }-{ left( frac { a-b }{ 2 } right) }^{ 2 } } }$
$=logleft| left( x-frac { a+b }{ 2 } right) +sqrt { (x-a)(x-b) } right| +c$

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

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:
$I=int { frac { x-2 }{ sqrt { 4-{ (x-2) }^{ 2 } } } dx+4int { frac { dx }{ sqrt { 4-{ (x-2) }^{ 2 } } } } }$

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) $letquad I=int { frac { dx }{ { x }^{ 2 }+2x+2 } } =int { frac { dx }{ (x+1)^{ 2 }+1 } }$
= (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) $int { frac { dx }{ sqrt { 9x-{ 4x }^{ 2 } } } } =frac { 1 }{ 2 } left[ frac { dx }{ sqrt { left( frac { 9 }{ 8 } right) ^{ 2 }-left[ { x }^{ 2 }-{ frac { 9 }{ 4 } }x+left( frac { 9 }{ 8 } right) ^{ 2 } right] } } right]$
$frac { 1 }{ 2 } { sin }^{ -1 }left( frac { 8x-9 }{ 9 } right) +c$

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NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4

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