NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 7
Chapter Name	Integrals
Exercise	Ex 7.3
Number of Questions Solved	24
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

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

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
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 3

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:
$\int { { (\frac { 1-cos2x }{ 2 } ) }^{ 2 }dx } \quad =\frac { 1 }{ 4 } \int { \left( { 1+cos }^{ 2 }2x-2cos2x \right) dx }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 10

Ex 7.3 Class 12 Maths Question 11.
cos⁴ 2x
Solution:
∫ cos⁴ 2x dx
$\int { { \left( \frac { 1+cos4x }{ 2 } \right) }^{ 2 } } dx$
tiwari academy class 12 maths Chapter 7 Integrals 11

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
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 15

Ex 7.3 Class 12 Maths Question 16.
tan⁴x
Solution:
let I = ∫tan⁴ dx
= ∫(sec²x-1)²dx
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 16

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

We hope the NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3 help you. If you have any query regarding NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3, drop a comment below and we will get back to you at the earliest.

NCERT Solutions
CBSE Sample Papers
NCERT Exemplar Solutions	LCM and GCF Calculator
TS Grewal Accountancy Class 12 Solutions
TS Grewal Accountancy Class 11 Solutions

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

Maths NCERT Solutions

SCIENCE NCERT SOLUTIONS

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.3

Reader Interactions

Leave a Reply Cancel reply

Footer

Maths NCERT Solutions

SCIENCE NCERT SOLUTIONS