NCERT Solutions for Class 12 Maths Chapter 7 Integrals

Table of Contents

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

The topics and sub-topics included in the Integrals chapter are the following:

Section Name	Topic Name
7	Integrals
7.1	Introduction
7.2	Integration as an Inverse Process of Differentiation
7.3	Methods of Integration
7.4	Integrals of some Particular Functions
7.5	Integration by Partial Fractions
7.6	Integration by Parts
7.7	Definite Integral
7.8	Fundamental Theorem of Calculus
7.9	Evaluation of Definite Integrals by Substitution
7.10	Some Properties of Definite Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals: Engineering aspirants and students appearing for CBSE Class 12 board exams must take the NCERT Mathematics textbooks seriously and finish them from top to bottom. They must also go through the NCERT solutions for Class 12 Maths and Class 11 Maths to have a better understanding of the various concepts. In this article, we will provide you with NCERT Solutions for Class 12 Maths Chapter 7 – Integrals which have been designed by the best teachers in India.

NCERT Solutions for Class 12 Maths Chapter 7 – Integrals contains step-by-step and detailed solutions for every question.

Introduction
Integration as an Inverse Process of Differentiation
Geometrical interpretation of indefinite integral
Some properties of indefinite integral
Comparison between differentiation and integration
Methods of Integration
Integration by substitution
Integration using trigonometric identities
Integrals of Some Particular Functions
Integration by Partial Fractions
Integration by Parts
Integral of the type
Integrals of some more types
Definite Integral
Definite integral as the limit of a sum
Fundamental Theorem of Calculus
Area function
First fundamental theorem of integral calculus
Second fundamental theorem of integral calculus
Evaluation of Definite Integrals by Substitution
Some Properties of Definite Integrals

Question 1.
sin 2x
Solution:

Question 2.
cos 3x
Solution:

Question 3.
${ e }^{ 2x }$
Solution:

Question 4.
(ax + c)²
Solution:

Question 5.
${ sin\quad 2x-4e }^{ 3x }$
Solution:

Find the following integrals in Exercises 6 to 20 :

Question 6.
$\int { \left( { 4e }^{ 3x }+1 \right) dx }$
Solution:

Question 7.
$\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx }$
Solution:

Question 8.
$\int { { (ax }^{ 2 }+bx+c)dx }$
Solution:

Question 9.
$\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx }$
Solution:

Question 10.
$\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx }$
Solution:

Question 11.
$\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx }$
Solution:

Question 12.
$\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx }$
Solution:

Question 13.
$\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx }$
Solution:

Question 14.
$\int { \left( 1-x \right) \sqrt { x } dx }$
Solution:

Question 15.
$\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx }$
Solution:

Question 16.
$\int { (2x-3cosx+{ e }^{ x })dx }$
Solution:

Question 17.
$\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx }$
Solution:

Question 18.
$\int { secx(secx+tanx)dx }$
Solution:

Question 19.
$\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx }$
Solution:

Question 20.
$\int { \frac { 2-3sinx }{ { cos }^{ 2 }x } dx }$
Solution:

Choose the correct answer in Exercises 21 and 22.
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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.2

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.2 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.2

Integrate the functions in Exercises 1 to 37:

Question 1.
$\frac { 2x }{ 1+{ x }^{ 2 } }$
Solution:

Question 2.
$\frac { { \left( logx \right) }^{ 2 } }{ x }$
Solution:

Question 3.
$\frac { 1 }{ x+xlogx }$
Solution:

Question 4.
sinx sin(cosx)
Solution:

Question 5.
sin(ax+b) cos(ax+b)
Solution:

Question 6.
$\sqrt { ax+b }$
Solution:

Question 7.
$x\sqrt { x+2 }$
Solution:

Question 8.
$x\sqrt { 1+{ 2x }^{ 2 } }$
Solution:

Question 9.
$(4x+2)\sqrt { { x }^{ 2 }+x+1 }$
Solution:

Question 10.
$\frac { 1 }{ x-\sqrt { x } }$
Solution:

Question 11.
$\frac { x }{ \sqrt { x+4 } } ,x>0$
Solution:

Question 12.
${ { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }$
Solution:

Question 13.
$\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } }$
Solution:

Question 14.
$\frac { 1 }{ x(logx)^{ m } } ,x>0$
Solution:

Question 15.
$\frac { x }{ 9-4{ x }^{ 2 } }$
Solution:

Question 16.
${ e }^{ 2x+3 }$
Solution:

Question 17.
$\frac { x }{ { e }^{ { x }^{ 2 } } }$
Solution:

Question 18.
$\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } }$
Solution:

Question 19.
$\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 }$
Solution:

Question 20.
$\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } }$
Solution:

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
= $\frac { 1 }{ 2 }t-x+c$
= $\frac { 1 }{ 2 }tan(2x-3)-x+c$
Question 22.
sec²(7-4x)
Solution:

Question 23.
$\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:

Question 24.
$\frac { 2cosx-3sinx }{ 6cosx+4sinx }$
Solution:

Question 25.
$\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } }$
Solution:

Question 26.
$\frac { cos\sqrt { x } }{ \sqrt { x } }$
Solution:

Question 27.
$\sqrt { sin2x } cos2x$
Solution:

Question 28.
$\frac { cosx }{ \sqrt { 1+sinx } }$
Solution:

Question 29.
cotx log sinx
Solution:

Question 30.
$\frac { sinx }{ 1+cosx }$
Solution:

Question 31.
$\frac { sinx }{ { (1+cosx) }^{ 2 } }$
Solution:

Question 32.
$\frac { 1 }{ 1+cotx }$
Solution:

Question 33.
$\frac { 1 }{ 1-tanx }$
Solution:

Question 34.
$\frac { \sqrt { tanx } }{ sinxcosx }$
Solution:

Question 35.
$\frac { { (1+logx) }^{ 2 } }{ x }$
Solution:

Question 36.
$\frac { (x+1){ (x+logx) }^{ 2 } }{ x }$
Solution:

Question 37.
$\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx$
Solution:

Choose the correct answer in exercises 38 and 39

Question 38.
$\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx }$
(a) 10x – x10 + C
(b) 10x + x10 + C
(c) (10x – x10) + C
(d) log (10x + x10) + C
Solution:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.3

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

NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.3

Find the integrals of the functions in Exercises 1 to 22.
Question 1.
sin²(2x+5)
Solution:

Question 2.
sin3x cos4x
Solution:

Question 3.
∫cos2x cos4x cos6x dx
Solution:

Question 4.
∫sin³(2x+1)dx
Solution:

Question 5.
sin³x cos³x
Solution:

Question 6.
sinx sin2x sin3x
Solution:

Question 7.
sin 4x sin 8x
Solution:

Question 8.
$\frac { 1-cosx }{ 1+cosx }$
Solution:

Question 9.
$\frac { cosx }{ 1+cosx }$
Solution:

Question 10.
∫sinx⁴ dx
Solution:

Question 11.
cos⁴ 2x
Solution:

Question 12.
$\frac { { sin }^{ 2 }x }{ 1+cosx }$
Solution:

Question 13.
$\frac { cos2x-cos2\alpha }{ cosx-cos\alpha }$
Solution:

Question 14.
$\frac { cosx-sinx }{ 1+sin2x }$
Solution:

Question 15.
$\int { { tan }^{ 3 }2x\quad sec2x\quad dx=I }$
Solution:

Question 16.
tan⁴x
Solution:

Question 17.
$\frac { { sin }^{ 3 }x+{ cos }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x }$
Solution:

Question 18.
$\frac { cos2x+{ 2sin }^{ 2 }x }{ { cos }^{ 2 }x }$
Solution:

Question 19.
$\frac { 1 }{ sinx{ cos }^{ 3 }x }$
Solution:

Question 20.
$\frac { cos2x }{ { (cosx+sinx) }^{ 2 } }$
Solution:

Question 21.
sin^-1 (cos x)
Solution:

Question 22.
$\int { \frac { 1 }{ cos(x-a)cos(x-b) } dx }$
Solution:

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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.4

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.4 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.4
Question 1.
$\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 }$
Solution:

Question 2.
$\frac { 1 }{ \sqrt { 1+{ 4x }^{ 2 } } }$
Solution:

Question 3.
$\frac { 1 }{ \sqrt { { (2-x) }^{ 2 }+1 } }$
Solution:

Question 4.
$\frac { 1 }{ \sqrt { 9-{ 25x }^{ 2 } } }$
Solution:

Question 5.
$\frac { 3x }{ 1+{ 2x }^{ 4 } }$
Solution:

Question 6.
$\frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } }$
Solution:

Question 7.
$\frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } }$
Solution:

Question 8.
$\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }+{ a }^{ 6 } } }$
Solution:

Question 9.
$\frac { { sec }^{ 2 }x }{ \sqrt { { tan }^{ 2 }x+4 } }$
Solution:

Question 10.
$\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } }$
Solution:

Question 11.
$\frac { 1 }{ { 9x }^{ 2 }+6x+5 }$
Solution:

Question 12.
$\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } }$
Solution:

Question 13.
$\frac { 1 }{ \sqrt { (x-1)(x-2) } }$
Solution:

Question 14.
$\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } }$
Solution:

Question 15.
$\frac { 1 }{ \sqrt { (x-a)(x-b) } }$
Solution:

Question 16.
$\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } }$
Solution:

Question 17.
$\frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } }$
Solution:

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= $-\frac { 11 }{ 3 }$

Question 19.
$\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } }$
Solution:

Question 20.
$\frac { x+2 }{ \sqrt { 4x-{ x }^{ 2 } } }$
Solution:

Question 21.
$\frac { x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } }$
Solution:

Question 22.
$\frac { x+3 }{ { x }^{ 2 }-2x-5 }$
Solution:

Question 23.
$\frac { 5x+3 }{ \sqrt { { x }^{ 2 }+4x+10 } }$
Solution:

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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.5

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.5 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.5

Question 1.
$\frac { x }{ (x+1)(x+2) }$
Solution:

Question 2.
$\frac { 1 }{ { x }^{ 2 }-9 }$
Solution:

Question 3.
$\frac { 3x-1 }{ (x-1)(x-2)(x-3) }$
Solution:

Question 4.
$\frac { x }{ (x-1)(x-2)(x-3) }$
Solution:

Question 5.
$\frac { 2x }{ { x }^{ 2 }+3x+2 }$
Solution:

Question 6.
$\frac { 1-{ x }^{ 2 } }{ x(1-2x) }$
Solution:

Question 7.
$\frac { x }{ \left( { x }^{ 2 }+1 \right) \left( x-1 \right) }$
Solution:

Question 8.
$\frac { x }{ { \left( x-1 \right) }^{ 2 }\left( x+2 \right) }$
Solution:

Question 9.
$\frac { 3x+5 }{ { x }^{ 3 }-{ x }^{ 2 }-x+1 }$
Solution:

Question 10.
$\frac { 2x-3 }{ ({ x }^{ 2 }-1)(2x+3) }$
Solution:

Question 11.
$\frac { 5x }{ (x-1)({ x }^{ 2 }-4) }$
Solution:

Question 12.
$\frac { { x }^{ 3 }+x+1 }{ { x }^{ 2 }-1 }$
Solution:

Question 13.
$\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) }$
Solution:

Question 14.
$\frac { 3x-1 }{ { (x+2) }^{ 2 } }$
Solution:

Question 15.
$\frac { 1 }{ { x }^{ 4 }-1 }$
Solution:

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

Question 17.
$\frac { cosx }{ (1-sinx)(2-sinx) }$
Solution:

Question 18.
$\frac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) }$
Solution:

Question 19.
$\frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) }$
Solution:

Question 20.
$\frac { 1 }{ x({ x }^{ 4 }-1) }$
Solution:

Question 21.
$\frac { 1 }{ { e }^{ x }-1 }$
Solution:

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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.6

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

Integrate the functions in Exercises 1 to 22.
Question 1.
x sinx
Solution:

Question 2.
x sin3x
Solution:

Question 3.
${ x }^{ 2 }{ e }^{ x }$
Solution:

Question 4.
x logx
Solution:

Question 5.
x log2x
Solution:

Question 6.
${ x }^{ 2 }logx$
Solution:

Question 7.
$x\quad { sin }^{ -1 }x$
Solution:

Question 8.
$x\quad { tan }^{ -1 }x$
Solution:

Question 9.
$x\quad { cos }^{ -1 }x$
Solution:

Question 10.
${ (sin }^{ -1 }{ x })^{ 2 }$
Solution:

Question 11.
$\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:

Question 12.
x sec²x
Solution:

Question 13.
${ ta }n^{ -1 }x$
Solution:

Question 14.
x(logx)²
Solution:

Question 15.
(x²+1)logx
Solution:

Question 16.
${ e }^{ x }(sinx+cosx)$
Solution:

Question 17.
$\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } }$
Solution:

Question 18.
$\frac { { e }^{ x }(1+sinx) }{ 1+cosx }$
Solution:

Question 19.
${ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right)$
Solution:

Question 20.
$\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } }$
Solution:

Question 21.
${ e }^{ 2x }sinx$
Solution:

Question 22.
${ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$
Solution:

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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.7

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.7 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.7

NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.7
Integral the function in exercises 1 to 9

Question 1.
$\sqrt { 4-{ x }^{ 2 } }$
Solution:

Question 2.
$\sqrt { 1-{ 4x }^{ 2 } }$
Solution:

Question 3.
$\sqrt { { x }^{ 2 }+4x+6 }$
Solution:

Question 4.
$\sqrt { { x }^{ 2 }+4x+1 }$
Solution:

Question 5.
$\sqrt { 1-4x-{ x }^{ 2 } }$
Solution:

Question 6.
$\sqrt { { x }^{ 2 }+4x-5 }$
Solution:

Question 7.
$\sqrt { 1+3x-{ x }^{ 2 } }$
Solution:

Question 8.
$\sqrt { { x }^{ 2 }+3x }$
Solution:

Question 9.
$\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } }$
Solution:

Choose the correct answer in the Exercises 10 to 11:

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:

Question 11.
$\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to$

Solution:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.8

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.8 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.8
Question 1.
$\int _{ a }^{ b }{ x\quad dx }$
Solution:

Question 2.
$\int _{ 0 }^{ 5 }{ (x+1)dx }$
Solution:

Question 3.
$\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx$
Solution:

Question 4.
$\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx$
Solution:

Question 5.
$\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad$
Solution:

Question 6.
$\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad$
Solution:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.9

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.9 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.9
Question 1.
$\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad$
Solution:

Question 2.
$\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx }$
Solution:

Question 3.
$\int _{ 1 }^{ 2 }{ \left( { 4x }^{ 3 }-{ 5x }^{ 2 }+6x+9 \right) dx }$
Solution:

Question 4.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx }$
Solution:

Question 5.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx }$
Solution:

Question 6.
$\int _{ 4 }^{ 5 }{ { e }^{ x }dx }$
Solution:

Question 7.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx }$
Solution:

Question 8.
$\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx }$
Solution:

Question 9.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } }$
Solution:

Question 10.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } }$
Solution:

Question 11.
$\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } }$
Solution:

Question 12.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx$
Solution:

Question 13.
$\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx$
Solution:

Question 14.
$\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx }$
Solution:

Question 15.
$\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx }$
Solution:

Question 16.
$\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx }$
Solution:

Question 17.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx }$
Solution:

Question 18.
$\int _{ 0 }^{ \pi }{ \left( { sin }^{ 2 }\frac { x }{ 2 } -{ cos }^{ 2 }\frac { x }{ 2 } \right) } dx$
Solution:

Question 19.
$\int _{ 0 }^{ 2 }{ \frac { 6x+3 }{ { x }^{ 2 }+4 } } dx$
Solution:

Question 20.
$\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx }$
Solution:

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:

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:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.10

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.10 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.10

Question 1:
Ex 7.10 Integrals Class 12 NCERT Solutions Ex 7.10 Q 1
Solution:
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 1
Question 2:

Solution:

Question 3:

Solution:

Question 4:
Integration Class 12 Ex 7.10 Q 4
Solution:

Question 5:

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Question 6:

Solution:
NCERT Class 12 Solutions Integrals Ex 7.10 Q 6

Question 7:

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Question 8:

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Question 9:

Solution:
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 9
Question 10:

Solution:

Question 11:
Integrals Class 12 Ex 7.10 11
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Question 12:

Solution:
NCERT Class 12 Solutions Ex 7.10 Q 12

Question 13:

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Question 14:
integrals class 12 ncert solutions Ex 7.10 Q 14
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Question 15:

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Question 16:
maths ncert solutions class 12 Ex 7.10 Q 16
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 16
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Question 18:

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Question 19:

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Question 20:

Solution:

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Integers Ex 7.11 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.11

Ex 7.11 Integrals Class 12 Maths NCERT Solutions Sol 1

integrals ncert solutions Ex 7.11 Q 5

Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 6

NCERT maths solutions class 12 Ex 7.11 Sol 11

integration class 12 ncert solutions Ex 7.11 Q 20
NCERT solutions for class 12 maths Ex 7.11 Sol 20

Maths NCERT Solution Class 12 Ex 7.11 Q 26

NCERT Solutions for Class 12 Maths Chapter 7 Integers Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 7 Integers Miscellaneous Exercise are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Integrals Miscellaneous Exercise

NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise
Question 1:

Solution: