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.

Class 12 Maths Chapter 7 NCERT Solutions Integrals

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

Integrals Ex 7.1 Class 12

Question 1.
sin 2x
Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 1
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:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Q 6
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:
Integrals Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 9
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:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 12
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:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.1 Q 14
Question 15.
$\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx }$
Solution:

Question 16.
$\int { (2x-3cosx+{ e }^{ x })dx }$
Solution:
Maths NCERT Class 12 Solutions Chapter 7 Integrals Ex 7.1 Q 16
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:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 19
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:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 21
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:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 22

Integrals Ex 7.2 Class 12

Integrate the functions in Exercises 1 to 37:

Question 1.
$\frac { 2x }{ 1+{ x }^{ 2 } }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Q 1
Question 2.
$\frac { { \left( logx \right) }^{ 2 } }{ x }$
Solution:

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

Question 4.
sinx sin(cosx)
Solution:
ncert solutions class 12 maths Chapter 7 Integrals Ex 7.2 Q 4
Question 5.
sin(ax+b) cos(ax+b)
Solution:
class 12 integration NCERT Solutions Chapter 7 Ex 7.2 Q 5
Question 6.
$\sqrt { ax+b }$
Solution:

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

Question 8.
$x\sqrt { 1+{ 2x }^{ 2 } }$
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 8

Question 9.
$(4x+2)\sqrt { { x }^{ 2 }+x+1 }$
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 9
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:
ncert class 12 maths solutions Integrals Ex 7.2 Q 12
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:
ncert maths solutions class 12 Integration Chapter 7 Ex 7.2 Q 16
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:
integration ncert solutions Class 12 chapter 7 Ex 7.2 Q 19
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:
maths class 12 ncert solutions Chapter 7 Integration Ex 7.2 Q 22
Question 23.
$\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } }$
Solution:

Question 24.
$\frac { 2cosx-3sinx }{ 6cosx+4sinx }$
Solution:
Ex 7.2 class 12 maths NCERT Solutions Q 24
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:
Integration Solutions class 12 Maths Chapter 7 Ex 7.2 Q 28
Question 29.
cotx log sinx
Solution:

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

Question 31.
$\frac { sinx }{ { (1+cosx) }^{ 2 } }$
Solution:
NCERT Maths Class 12 Solutions Chapter 7 Ex 7.2 Q 31
Question 32.
$\frac { 1 }{ 1+cotx }$
Solution:

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

Question 34.
$\frac { \sqrt { tanx } }{ sinxcosx }$
Solution:
NCERT Integration Solutions Ex 7.2 Q 34
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:
NCERT Integration Solutions Ex 7.2 Q 38
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 integrals Ex 7.2 Q 39

Integrals Ex 7.3 Class 12

Find the integrals of the functions in Exercises 1 to 22.
Question 1.
sin²(2x+5)
Solution:
Ex 7.3 Class 12 Maths NCERT Solutions Q 1
Question 2.
sin3x cos4x
Solution:

Question 3.
∫cos2x cos4x cos6x dx
Solution:

Question 4.
∫sin³(2x+1)dx
Solution:

Question 5.
sin³x cos³x
Solution:
NCERT Solutions Integration Chapter 7 Ex 7.3 Q 5
Question 6.
sinx sin2x sin3x
Solution:

Question 7.
sin 4x sin 8x
Solution:

Question 8.
$\frac { 1-cosx }{ 1+cosx }$
Solution:
chapter 7 maths class 12 NCERT Solutions Ex 7.3 Q 8
Question 9.
$\frac { cosx }{ 1+cosx }$
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 9
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:
Chapter 7 Class 12 Maths NCERT Solutions Ex 7.3 Q 14
Question 15.
$\int { { tan }^{ 3 }2x\quad sec2x\quad dx=I }$
Solution:

Question 16.
tan⁴x
Solution:
NCERT Solutions for Class 12 Maths Ex 7.3 Q 16
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:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.3 Q 20
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:
NCERT Solution Class 12 Maths Ex 7.3 Q 23
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:

Integrals Class 12 Ex 7.4

Question 1.
$\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 }$
Solution:
Class 12 Integrals NCERT Solutions Ex 7.4 Q 1
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:
NCERT Solutions for Class 12 Maths Chapter 7 PDF Ex 7.4 Q 5
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:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 9
Question 10.
$\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } }$
Solution:

Question 11.
$\frac { 1 }{ { 9x }^{ 2 }+6x+5 }$
Solution:
miscellaneous exercise on chapter 7 class 12 Q 11
Question 12.
$\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } }$
Solution:

Question 13.
$\frac { 1 }{ \sqrt { (x-1)(x-2) } }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 13
Question 14.
$\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } }$
Solution:
miscellaneous exercise on chapter 7 class 12 Q 14
Question 15.
$\frac { 1 }{ \sqrt { (x-a)(x-b) } }$
Solution:

Question 16.
$\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } }$
Solution:
Class 12 Maths Solutions Ex 7.4 Q 16
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= $-\frac { 11 }{ 3 }$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 18

Question 19.
$\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } }$
Solution:
Class 12 Maths Solutions Ex 7.4 Q 18

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:
NCERT Maths Class 12 Ex 7.4 Q 21

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 Maths Class 12 Ex 7.4 Q 24

Integrals Class 12 Ex 7.5

Question 1.
$\frac { x }{ (x+1)(x+2) }$
Solution:
NCERT Solutions Maths Class 12 Ex 7.5 Q 1
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:
Chapter 7 Class 12 Maths Ex 7.5 Q 5
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:
Class 12 Integration NCERT Solutions Ex 7.5 Q 8
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:
Class 12 Integration NCERT Solutions Ex 7.5 Q 12
Question 13.
$\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Q 13
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:
Class 12 Maths Ex 7.5 Q 16
Class 12 Maths Ex 7.5 Q 16 - i
Question 17.
$\frac { cosx }{ (1-sinx)(2-sinx) }$
Solution:
Class 12 Maths Ex 7.5 Q 17
Question 18.
$\frac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) }$
Solution:
Class 12 Maths Ex 7.5 Q 18
Class 12 Maths Ex 7.5 Q 18 - i
Question 19.
$\frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) }$
Solution:
NCERT Class 12 Maths Ex 7.5 Q 19
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:
NCERT Class 12 Maths Ex 7.5 Q 22
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 Class 12 Maths Ex 7.5 Q 23

Integrals Class 12 Ex 7.6

Question 1.
x sinx
Solution:
Maths Class 12 NCERT Solutions Chapter 7 Ex 7.6 Q 1
Question 2.
x sin3x
Solution:

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

Question 4.
x logx
Solution:

Question 5.
x log2x
Solution:
NCERT Maths Solution Class 12 Ex 7.6 Q 5
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:
Class 12 Maths NCERT Solution Ex 7.6 Q 9

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:
Class 12 Maths NCERT Solution Q 14
Question 15.
(x²+1)logx
Solution:

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

Question 17.
$\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } }$
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 17
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:
Ch 7 Maths Class 12 Ex 7.6 Q 20
Question 21.
${ e }^{ 2x }sinx$
Solution:

Question 22.
${ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right)$
Solution:
Maths Class 12 Chapter 7 NCERT Solutions Ex 7.6 Q 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:
Maths Class 12 Chapter 7 NCERT Solutions Ex 7.6 Q 23
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 Maths Solutions for Class 12 Ex 7.6 Q 24

Class 12 Integrals Ex 7.7

Integral the function in exercises 1 to 9

Question 1.
$\sqrt { 4-{ x }^{ 2 } }$
Solution:
Maths NCERT Class 12 Ex 7.7 Q 1
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:
NCERT Solutions Class 12 Ex 7.7 Q 6
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:
Exercise 7.7 Class 12 Maths Q 10
Question 11.
$\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to$
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 11
Solution:

Class 12 Integrals Ex 7.8

Question 1.
$\int _{ a }^{ b }{ x\quad dx }$
Solution:
Maths NCERT Class 12 Ex 7.8 Q 1

Question 2.
$\int _{ 0 }^{ 5 }{ (x+1)dx }$
Solution:
Maths NCERT Class 12 Ex 7.8 Q 2

Question 3.
$\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx$
Solution:
Maths NCERT Class 12 Ex 7.8 Q 3

Question 4.
$\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx$
Solution:
NCERT Solutions Class 12 Ex 7.8 Q 4

Question 5.
$\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad$
Solution:
Maths NCERT Class 12 Ex 7.8 Q 5

Question 6.
$\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad$
Solution:
Maths NCERT Class 12 Ex 7.8 Q 6

Class 12 Integrals Ex 7.9

Question 1.
$\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 1

Question 2.
$\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 2

Question 3.
$\int _{ 1 }^{ 2 }{ \left( { 4x }^{ 3 }-{ 5x }^{ 2 }+6x+9 \right) dx }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 3

Question 4.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx }$
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 4

Question 5.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx }$
Solution:
NCERT Integration Ex 7.9 Q 5

Question 6.
$\int _{ 4 }^{ 5 }{ { e }^{ x }dx }$
Solution:
NCERT Integration Ex 7.9 Q 6

Question 7.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx }$
Solution:
NCERT Integration Ex 7.9 Q 7

Question 8.
$\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx }$
Solution:
NCERT Integration Ex 7.9 Q 8

Question 9.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } }$
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 9

Question 10.
$\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } }$
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 10

Question 11.
$\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } }$
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 11

Question 12.
$\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx$
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 12

Question 13.
$\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx$
Solution:
Ex 7.9 Class 12 Q 13

Question 14.
$\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx }$
Solution:
Ex 7.9 Class 12 Q 14

Question 15.
$\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx }$
Solution:
Ex 7.9 Class 12 Q 15

Question 16.
$\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx }$
Solution:
Ex 7.9 Class 12 Q 16
Ex 7.9 Class 12 Q 16 - i

Question 17.
$\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx }$
Solution:
Ex 7.9 Class 12 Q 17

Question 18.
$\int _{ 0 }^{ \pi }{ \left( { sin }^{ 2 }\frac { x }{ 2 } -{ cos }^{ 2 }\frac { x }{ 2 } \right) } dx$
Solution:
Maths NCERT Class 12 Solutions Ex 7.9 Q 18

Question 19.
$\int _{ 0 }^{ 2 }{ \frac { 6x+3 }{ { x }^{ 2 }+4 } } dx$
Solution:
Maths NCERT Class 12 Solutions Ex 7.9 Q 19

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

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:
Maths NCERT Class 12 Solutions Ex 7.9 Q 21

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:
Maths NCERT Class 12 Solutions Ex 7.9 Q 22

Integration Class 12 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:

Solution:

Question 6:

Solution:
NCERT Class 12 Solutions Integrals Ex 7.10 Q 6

Question 7:

Solution:

Question 8:
Integrals Class 12 Ex 7.10 8
Solution:

Question 9:

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

Question 11:
Integrals Class 12 Ex 7.10 11
Solution:

Question 12:
integrals class 12 ncert solutions Ex 7.10 Q 12
Solution:
NCERT Class 12 Solutions Ex 7.10 Q 12

Question 13:

Solution:

Question 14:

Solution:

Question 15:

Solution:

Question 16:
maths ncert solutions class 12 Ex 7.10 Q 16
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 16
Question 17:

Solution:

Question 18:

Solution:

Question 19:

Solution:

Question 20:

Solution:

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

class 12 maths ncert solutions Ex 7.11 Q 1
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 12

NCERT solutions for class 12 maths Ex 7.11 Sol 16

Maths NCERT Solution Class 12 Ex 7.11 Q 26

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Question 1:
ncert solutions class 12 maths Miscellaneous Questions Q 1
Solution:

Question 2:
ncert solutions class 12 maths Miscellaneous Questions Q 2
Solution:

Question 3:
miscellaneous exercise on chapter 7 class 12 Q 3
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 3 - i

Question 4:
ncert solutions class 12 maths Miscellaneous Questions Q 4
Solution:
miscellaneous exercise on chapter 7 class 12 Q 4 - i

Question 5:
ncert solutions class 12 maths Miscellaneous Questions Q 5
Solution:

Question 6:
ncert solutions class 12 maths Miscellaneous Questions Q 6
Solution:

Question 7:
ncert solutions class 12 maths Miscellaneous Question Q 7
Solution:

Question 8:
ncert solutions class 12 maths Miscellaneous Questions Q 8
Solution:

Question 9:
ncert solutions for class 12 maths chapter 7 miscellaneous exercise Q 9
Solution:

Question 10:
ncert solutions class 12 maths Miscellaneous Questions Q 10
Solution:

Question 11:

Solution:

Question 12:
ncert solutions class 12 maths Miscellaneous Questions Q 12
Solution:
ncert miscellaneous solutions class 12 Q 12 - i

Question 13:
ncert solutions class 12 maths Miscellaneous Questions Q 13
Solution:

Question 14:
ncert solutions class 12 maths Miscellaneous Questions Q 14
Solution:
ncert solutions for class 12 maths chapter 7 miscellaneous Q 14

Question 15:
ncert solutions class 12 maths Miscellaneous Questions Q 15
Solution:

Question 16:
ncert integrals miscellaneous solutions Q 16
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 16 - i

Question 17:
ncert solutions class 12 maths Miscellaneous Questions Q 17
Solution:

Question 18:
ncert solutions class 12 maths Miscellaneous Questions Q 18
Solution:

Question 20:
ncert solutions class 12 maths Miscellaneous Questions Q 20
Solution:

Question 21:
ncert solutions class 12 maths Miscellaneous Questions Q 21
Solution:

Question 22:
ncert solutions class 12 maths Miscellaneous Questions Q 22
Solution:

Question 23:
ncert solutions class 12 maths Miscellaneous Questions Q 23
Solution:

Question 24:
ncert solutions class 12 maths Miscellaneous Questions Q 24
Solution:

ncert miscellaneous solutions class 12 Q 24 - ii

Question 25:
ncert solutions class 12 maths Miscellaneous Questions Q 25
Solution:

Question 26:
ncert solutions class 12 maths Miscellaneous Questions Q 26
Solution:

Question 27:
maths class 12 ncert solutions miscellaneous exercise 27
Solution:

Question 28:
maths class 12 ncert solutions miscellaneous exercise 28
Solution:

Question 29:
maths class 12 ncert solutions miscellaneous exercise 29
Solution:

Question 30:
maths class 12 ncert solutions miscellaneous exercise 30

Question 31:
maths class 12 ncert solutions miscellaneous exercise 31
Solution:

Question 32:
maths class 12 ncert solutions miscellaneous exercise 32
Solution:

Question 33:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 33
Solution:

Question 34:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 34
Solution:

Question 35:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 35
Solution:

Question 36:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 36
Solution:

Question 37:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 37
Solution:

Question 38:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 38
Solution:

Question 39:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 39
Solution:

Question 40:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 40
Solution:

Question 41:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 41
Solution:

Question 42:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 42
Solution:

Question 43:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 43
Solution:

Question 44:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 44
Solution:

Class 12 Maths NCERT Solutions

Updated NCERT Solutions for Class 12 Maths for 2019 Exams.

The post NCERT Solutions for Class 12 Maths Chapter 7 Integrals appeared first on Learn CBSE.

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

Class 12 Maths Chapter 7 NCERT Solutions Integrals

Integrals Ex 7.1 Class 12

Integrals Ex 7.2 Class 12

Integrals Ex 7.3 Class 12

Integrals Class 12 Ex 7.4

Integrals Class 12 Ex 7.5

Integrals Class 12 Ex 7.6

Class 12 Integrals Ex 7.7

Class 12 Integrals Ex 7.8

Class 12 Integrals Ex 7.9

Integration Class 12 Ex 7.10

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Class 12 Maths NCERT Solutions

Trending Articles

Practice Sheet of Right form of verbs for HSC Students

Download: FK ft Shenky – Nakuyewa ”Prod by: Shenky”

How to win at Markstrat (Markstrat Tips and Tricks) – Vodites

Ominde Commission Report and Recommendations – Ominde Report of 1964

Bureau of Internal Revenue: Regional Offices (Directory)

GO 53 on Enhancement of Ex-gratia upto 5 Lakhs Toddy Tappers in Telangana

Cakewalk CA-2A Leveling Amplifier v2.0.1.97 WiN, v2.0.1.96 OSX Incl Keygen

Mp3 Download: Mdu - Kunjenjenjena

How the kill the job , when DTP request running for long hours.

Microsoft Intune から展開しているアプリのアップデートについて

18-year-old girl was beaten for half an hour by two Northampton men in 'an...

Car crash in Dunton Bassett leaves driver in critical condition

Macky 2, Two Others In Road Accident

Application log 00000000000000089514: Could not convert queue DLVST90CLNT

Detroit mafia: D’Anna Brothers agree to plea deal

Delivery block field greyed out using VA02

Muloraki Au

【個人撮影】スマホのプライベート映像♪「中に出さないで///」カラオケ屋での生ハメ撮りが流出ｗ【リベンジポルノ】＠PornHub

BREAKING NEWS: Diamond Platnumz Is Reported Dead After Ghastly Car Accident

FIAT 500 B0111 B0112