NCERT Solutions for Class 12 Maths Chapter 7 Integrals

August 21, 2019 by phani 45 Comments

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.

Free download NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.11, Ex 7.10, Ex 7.9, Ex 7.8, Ex 7.7, Ex 7.6, Ex 7.5, Ex 7.4, Ex 7.3, Ex 7.2, Ex 7.1 Integrals PDF in Hindi Medium as well as in English Medium for CBSE, Uttarakhand, Bihar, MP Board, Gujarat Board, BIE, Intermediate and UP Board students, who are using NCERT Books based on updated CBSE Syllabus for the session 2019-20.

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:

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:

Move to Top → Class 12 Integrals Chapter 7

Integrals Ex 7.2 Class 12

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:

Move to Top → Class 12 Integrals Chapter 7

Integrals Ex 7.3 Class 12

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:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 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:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 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:

Move to Top → Class 12 Integrals Chapter 7

Integrals Class 12 Ex 7.6

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:

Move to Top → Class 12 Integrals Chapter 7

Class 12 Integrals 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:

Move to Top → Class 12 Integrals Chapter 7

Class 12 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:

Move to Top → Class 12 Integrals Chapter 7

Class 12 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:

Move to Top → Class 12 Integrals Chapter 7

Integration Class 12 Ex 7.10

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Move to Top → Class 12 Integrals Chapter 7

NCERT Solutions for Class 2 Maths Integration Class 12 Ex 7.11

Move to Top → Class 12 Integrals Chapter 7

NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

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Move to Top → Class 12 Integrals Chapter 7

Integration Formulas

Move to Top → Class 12 Integrals Chapter 7

गणित कक्षा 12 समाकलन प्रश्नावली 7.1

निम्नलिखित फलनों के प्रतिअवकलज ( समाकलन) निरीक्षण विधि द्वारा ज्ञात कीजिए।

प्रश्न 1.
sin 2x
हल-
∫sin 2x dx
हम जानते हैं कि, $\frac { d }{ dx }$ cos2x = -2 sin 2x

प्रश्न 2.
cos 3x
हल-
∫cos 3x dx
हम जानते हैं कि, $\frac { d }{ dx }$ sin 3x = 3 cos 3x

प्रश्न 3.
e^2x
हल-
∫e^2x dx
हम जानते हैं कि,

प्रश्न 4.
(ax + b)²
हल-
∫(ax + b)² dx
हम जानते हैं कि,

प्रश्न 5.
sin 2x – 4e^3x
हल-
∫(sin 2x -4e^3x) dx = ∫sin 2x dx – 4∫e^3x …(1)

निम्नलिखित समाकलनों को ज्ञात कीजिए।

प्रश्न 6.
∫(4e^3x + 1) dx
हल-
∫(4e^3x + 1) dx = 4∫e^3xdx + ∫1 dx

प्रश्न 7.

हल-

प्रश्न 8.
∫(ax² + bx + c) dx
हल-
∫(ax² + bx + c) dx = a∫x²dx + b∫xdx + c∫1 dx

प्रश्न 9.
∫(2x² + e^x) dx
हल-
∫(2x² + e^x) dx = 2∫x² dx + ∫e^x dx

प्रश्न 10.

हल-

प्रश्न 11.

हल-

प्रश्न 12.

हल-

प्रश्न 13.

हल-

प्रश्न 14.
∫(1-x)√x dx
हल-
∫(1-x)√x dx
=∫(√x-x√x)dx

प्रश्न 15.
∫√x(3x² + 2x + 3)dx
हल-
∫√x(3x² + 2x + 3)dx

प्रश्न 16.
∫(2x – 3cosx + e^x)dx
हल-
∫(2x – 3cosx + e^x)dx

प्रश्न 17.
∫(2x² – 3sinx + 5√x)dx
हल-
∫(2x² – 3sinx + 5√x)dx
= 2∫x² dx – 3 ∫ sinx dx + 5∫√x dx

प्रश्न 18.
∫secx (sec x + tan x) dx
हल-
∫sec x (sec x + tan x) dx
= ∫(sec² x + sec x tan x) dx
= ∫(sec² x dx
= ∫sec x tan x dx
tan x + sec x + C

प्रश्न 19.

हल-

प्रश्न 20.

हल-

प्रश्न 21 एवं 22 में सही उत्तर का चयन कीजिए

प्रश्न 21

हल-

प्रश्न 22

हल-

Move to Top → Class 12 Integrals Chapter 7

Class 12 Maths NCERT Solutions

Updated NCERT Solutions for Class 12 Maths for 2019 Exams.

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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

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Integrals Class 12 Ex 7.4

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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

Integration Formulas

गणित कक्षा 12 समाकलन प्रश्नावली 7.1

Class 12 Maths NCERT Solutions

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