( NEW SYLLABUS 2021) NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals

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

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

  1. Introduction
  2. Integration as an Inverse Process of Differentiation
  3. Geometrical interpretation of indefinite integral
  4. Some properties of indefinite integral
  5. Comparison between differentiation and integration
  6. Methods of Integration
  7. Integration by substitution
  8. Integration using trigonometric identities
  9. Integrals of Some Particular Functions
  10. Integration by Partial Fractions
  11. Integration by Parts
  12. Integral of the type
  13. Integrals of some more types
  14. Definite Integral
  15. Definite integral as the limit of a sum
  16. Fundamental Theorem of Calculus
  17. Area function
  18. First fundamental theorem of integral calculus
  19. Second fundamental theorem of integral calculus
  20. Evaluation of Definite Integrals by Substitution
  21. Some Properties of Definite Integrals

Integrals Ex 7.1 Class 12

Ex 7.1 Class 12 Maths Question 1.
sin 2x
Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 1
Ex 7.1 Class 12 Maths Question 2.
cos 3x
Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 2
Ex 7.1 Class 12 Maths Question 3.
[latex s=2]{ e }^{ 2x }[/latex]
Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 3
Ex 7.1 Class 12 Maths Question 4.
(ax + c)²
Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 4
Ex 7.1 Class 12 Maths Question 5.
[latex s=2]{ sin\quad 2x-4e }^{ 3x }[/latex]
Ex 7.1 Class 12 Maths Solution:
Class 12 Integrals NCERT Solutions Ex 7.1 Q 5
Find the following integrals in Exercises 6 to 20 :

Ex 7.1 Class 12 Maths Question 6.
[latex s=2]\int { \left( { 4e }^{ 3x }+1 \right) dx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Q 6
Ex 7.1 Class 12 Maths Question 7.
[latex s=2]\int { { x }^{ 2 }\left( 1-\frac { 1 }{ { x }^{ 2 } } \right) dx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Q 7
Ex 7.1 Class 12 Maths Question 8.
[latex s=2]\int { { (ax }^{ 2 }+bx+c)dx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.1 Q 8
Ex 7.1 Class 12 Maths Question 9.
[latex s=2]\int { \left( { 2x }^{ 2 }+{ e }^{ x } \right) dx } [/latex]
Solution:
Integrals Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 9
Ex 7.1 Class 12 Maths Question 10.
[latex s=2]\int { { \left[ \sqrt { x } -\frac { 1 }{ \sqrt { x } } \right] }^{ 2 }dx } [/latex]
Solution:
Integrals Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 10
Ex 7.1 Class 12 Maths Question 11.
[latex s=2]\int { \frac { { x }^{ 3 }+{ 5x }^{ 2 }-4 }{ { x }^{ 2 } } dx } [/latex]
Solution:
Integrals Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 11
Ex 7.1 Class 12 Maths Question 12.
[latex s=2]\int { \frac { { x }^{ 3 }+3x+4 }{ \sqrt { x } } dx } [/latex]
Solution:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 12
Ex 7.1 Class 12 Maths Question 13.
[latex s=2]\int { \frac { { x }^{ 3 }-{ x }^{ 2 }+x-1 }{ x-1 } dx } [/latex]
Solution:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 13
Ex 7.1 Class 12 Maths Question 14.
[latex s=2]\int { \left( 1-x \right) \sqrt { x } dx } [/latex]
Solution:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.1 Q 14
Ex 7.1 Class 12 Maths Question 15.
[latex s=2]\int { \sqrt { x } \left( { 3x }^{ 2 }+2x+3 \right) dx } [/latex]
Solution:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.1 Q 15
Ex 7.1 Class 12 Maths Question 16.
[latex s=2]\int { (2x-3cosx+{ e }^{ x })dx } [/latex]
Solution:
Maths NCERT Class 12 Solutions Chapter 7 Integrals Ex 7.1 Q 16
Ex 7.1 Class 12 Maths Question 17.
[latex s=2]\int { \left( { 2x }^{ 2 }-3sinx+5\sqrt { x } \right) dx } [/latex]
Solution:
Maths NCERT Class 12 Solutions Chapter 7 Integrals Ex 7.1 Q 17
Ex 7.1 Class 12 Maths Question 18.
[latex s=2]\int { secx(secx+tanx)dx } [/latex]
Solution:
Maths NCERT Class 12 Solutions Chapter 7 Integrals Ex 7.1 Q 18
Ex 7.1 Class 12 Maths Question 19.
[latex s=2]\int { \frac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } dx } [/latex]
Solution:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 19
Ex 7.1 Class 12 Maths Question 20.
[latex s=2]\int { \frac { 2-3sinx }{ { cos }^{ 2 }x } dx } [/latex]
Solution:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 20

Choose the correct answer in Exercises 21 and 22.
Ex 7.1 Class 12 Maths Question 21.
The antiderivative [latex s=2]\left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right) [/latex] equals
(a) [latex s=2]\frac { 1 }{ 3 } { x }^{ \frac { 1 }{ 3 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c[/latex]
(b) [latex s=2]\frac { 2 }{ 3 } { x }^{ \frac { 2 }{ 3 } }+{ \frac { 1 }{ 2 } x }^{ 2 }+c[/latex]
(c) [latex s=2]\frac { 2 }{ 3 } { x }^{ \frac { 3 }{ 2 } }+{ 2x }^{ \frac { 1 }{ 2 } }+c[/latex]
(d) [latex s=2]\frac { 3 }{ 2 } { x }^{ \frac { 3 }{ 2 } }+\frac { 1 }{ 2 } { x }^{ \frac { 1 }{ 2 } }+c[/latex]
Solution:
Integration Class 12 NCERT Solutions Chapter 7 Ex 7.1 Q 21
Ex 7.1 Class 12 Maths Question 22.
If [latex s=2]\frac { d }{ dx } f(x)={ 4x }^{ 3 }-\frac { 3 }{ { x }^{ 4 } } [/latex] such that f(2)=0 then f(x) is
(a) [latex s=2]{ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } -\frac { 129 }{ 8 } [/latex]
(b) [latex s=2]{ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } +\frac { 129 }{ 8 } [/latex]
(c) [latex s=2]{ x }^{ 4 }+\frac { 1 }{ { x }^{ 3 } } +\frac { 129 }{ 8 } [/latex]
(d) [latex s=2]{ x }^{ 3 }+\frac { 1 }{ { x }^{ 4 } } -\frac { 129 }{ 8 } [/latex]
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:

Ex 7.2 Class 12 Maths Question 1.
[latex s=2]\frac { 2x }{ 1+{ x }^{ 2 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Q 1
Ex 7.2 Class 12 Maths Question 2.
[latex s=2]\frac { { \left( logx \right) }^{ 2 } }{ x } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Q 2
Ex 7.2 Class 12 Maths Question 3.
[latex s=2]\frac { 1 }{ x+xlogx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.2 Q 3
Ex 7.2 Class 12 Maths Question 4.
sinx sin(cosx)
Solution:
ncert solutions class 12 maths Chapter 7 Integrals Ex 7.2 Q 4
Ex 7.2 Class 12 Maths Question 5.
sin(ax+b) cos(ax+b)
Solution:
class 12 integration NCERT Solutions Chapter 7 Ex 7.2 Q 5
Ex 7.2 Class 12 Maths Question 6.
[latex s=2]\sqrt { ax+b } [/latex]
Solution:
class 12 integration NCERT Solutions Chapter 7 Ex 7.2 Q 6
Ex 7.2 Class 12 Maths Question 7.
[latex s=2]x\sqrt { x+2 } [/latex]
Solution:
class 12 integration NCERT Solutions Chapter 7 Ex 7.2 Q 7
Ex 7.2 Class 12 Maths Question 8.
[latex s=2]x\sqrt { 1+{ 2x }^{ 2 } } [/latex]
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 8

Ex 7.2 Class 12 Maths Question 9.
[latex s=2](4x+2)\sqrt { { x }^{ 2 }+x+1 } [/latex]
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 9
Ex 7.2 Class 12 Maths Question 10.
[latex s=2]\frac { 1 }{ x-\sqrt { x } } [/latex]
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 10
Ex 7.2 Class 12 Maths Question 11.
[latex s=2]\frac { x }{ \sqrt { x+4 } } ,x>0[/latex]
Solution:
class 12 maths integration NCERT Solutions Chapter 7 Ex 7.2 Q 11
Ex 7.2 Class 12 Maths Question 12.
[latex s=2]{ { (x }^{ 3 }-1) }^{ \frac { 1 }{ 3 } }.{ x }^{ 5 }[/latex]
Solution:
ncert class 12 maths solutions Integrals Ex 7.2 Q 12
Ex 7.2 Class 12 Maths Question 13.
[latex s=2]\frac { { x }^{ 2 } }{ { { (2+3x }^{ 3 }) }^{ 3 } } [/latex]
Solution:
ncert class 12 maths solutions Integrals Ex 7.2 Q 13
Ex 7.2 Class 12 Maths Question 14.
[latex s=2]\frac { 1 }{ x(logx)^{ m } } ,x>0[/latex]
Solution:
ncert class 12 maths solutions Integrals Ex 7.2 Q 14
Ex 7.2 Class 12 Maths Question 15.
[latex s=2]\frac { x }{ 9-4{ x }^{ 2 } } [/latex]
Solution:
ncert class 12 maths solutions Integrals Ex 7.2 Q 14
Ex 7.2 Class 12 Maths Question 16.
[latex s=2]{ e }^{ 2x+3 }[/latex]
Solution:
ncert maths solutions class 12 Integration Chapter 7 Ex 7.2 Q 16
Ex 7.2 Class 12 Maths Question 17.
[latex s=2]\frac { x }{ { e }^{ { x }^{ 2 } } } [/latex]
Solution:
ncert maths solutions class 12 Integration Chapter 7 Ex 7.2 Q 17
Ex 7.2 Class 12 Maths Question 18.
[latex s=2]\frac { { e }^{ { tan }^{ -1 }x } }{ 1+{ x }^{ 2 } } [/latex]
Solution:
ncert maths solutions class 12 Integration Chapter 7 Ex 7.2 Q 18
Ex 7.2 Class 12 Maths Question 19.
[latex s=2]\frac { { e }^{ 2x }-1 }{ { e }^{ 2x }+1 } [/latex]
Solution:
integration ncert solutions Class 12 chapter 7 Ex 7.2 Q 19
Ex 7.2 Class 12 Maths Question 20.
[latex s=2]\frac { { e }^{ 2x }-{ e }^{ 2x } }{ { e }^{ 2x }+{ e }^{ -2x } } [/latex]
Solution:
integration ncert solutions Class 12 chapter 7 Ex 7.2 Q 20
Ex 7.2 Class 12 Maths 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 = [latex s=2]\frac { 1 }{ 2 }[/latex] ∫sec²t dt-x+c
= [latex s=2]\frac { 1 }{ 2 }t-x+c[/latex]
= [latex s=2]\frac { 1 }{ 2 }tan(2x-3)-x+c[/latex]
Ex 7.2 Class 12 Maths Question 22.
sec²(7-4x)
Solution:
maths class 12 ncert solutions Chapter 7 Integration Ex 7.2 Q 22
Ex 7.2 Class 12 Maths Question 23.
[latex s=2]\frac { { sin }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } [/latex]
Solution:
maths class 12 ncert solutions Chapter 7 Integration Ex 7.2 Q 23
Ex 7.2 Class 12 Maths Question 24.
[latex s=2]\frac { 2cosx-3sinx }{ 6cosx+4sinx }[/latex]
Solution:
Ex 7.2 class 12 maths NCERT Solutions Q 24
Ex 7.2 Class 12 Maths Question 25.
[latex s=2]\frac { 1 }{ { cos }^{ 2 }x{ (1-tanx) }^{ 2 } } [/latex]
Solution:
Ex 7.2 class 12 maths NCERT Solutions Q 25
Ex 7.2 Class 12 Maths Question 26.
[latex s=2]\frac { cos\sqrt { x } }{ \sqrt { x } } [/latex]
Solution:
Ex 7.2 class 12 maths NCERT Solutions Q 26
Ex 7.2 Class 12 Maths Question 27.
[latex s=2]\sqrt { sin2x } cos2x[/latex]
Solution:
Ex 7.2 class 12 maths NCERT Solutions Q 27
Ex 7.2 Class 12 Maths Question 28.
[latex s=2]\frac { cosx }{ \sqrt { 1+sinx } } [/latex]
Solution:
Integration Solutions class 12 Maths Chapter 7 Ex 7.2 Q 28
Ex 7.2 Class 12 Maths Question 29.
cotx log sinx
Solution:
Integration Solutions class 12 Maths Chapter 7 Ex 7.2 Q 29
Ex 7.2 Class 12 Maths Question 30.
[latex s=2]\frac { sinx }{ 1+cosx }[/latex]
Solution:
Integration Solutions class 12 Maths Chapter 7 Ex 7.2 Q 30
Ex 7.2 Class 12 Maths Question 31.
[latex s=2]\frac { sinx }{ { (1+cosx) }^{ 2 } } [/latex]
Solution:
NCERT Maths Class 12 Solutions Chapter 7 Ex 7.2 Q 31
Ex 7.2 Class 12 Maths Question 32.
[latex s=2]\frac { 1 }{ 1+cotx }[/latex]
Solution:
NCERT Maths Class 12 Solutions Chapter 7 Ex 7.2 Q 32
Ex 7.2 Class 12 Maths Question 33.
[latex s=2]\frac { 1 }{ 1-tanx }[/latex]
Solution:
NCERT Maths Class 12 Solutions Chapter 7 Ex 7.2 Q 33
Ex 7.2 Class 12 Maths Question 34.
[latex s=2]\frac { \sqrt { tanx } }{ sinxcosx } [/latex]
Solution:
NCERT Integration Solutions Ex 7.2 Q 34
Ex 7.2 Class 12 Maths Question 35.
[latex s=2]\frac { { (1+logx) }^{ 2 } }{ x } [/latex]
Solution:
NCERT Integration Solutions Ex 7.2 Q 35
Ex 7.2 Class 12 Maths Question 36.
[latex s=2]\frac { (x+1){ (x+logx) }^{ 2 } }{ x } [/latex]
Solution:
NCERT Integration Solutions Ex 7.2 Q 36
Ex 7.2 Class 12 Maths Question 37.
[latex s=2]\frac { { x }^{ 3 }sin({ tan }^{ -1 }{ x }^{ 4 }) }{ 1+{ x }^{ 8 } } dx [/latex]
Solution:
NCERT Integration Solutions Ex 7.2 Q 37
Choose the correct answer in exercises 38 and 39

Ex 7.2 Class 12 Maths Question 38.
[latex s=2]\int { \frac { { 10x }^{ 9 }+{ 10 }^{ x }log{ e }^{ 10 } }{ { x }^{ 10 }+{ 10 }^{ x } } dx } [/latex]
(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
Ex 7.2 Class 12 Maths Question 39.
[latex s=2]\int { \frac { dx }{ { sin }^{ 2 }x{ \quad cos }^{ 2 }x } = } [/latex]
(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.
Ex 7.3 Class 12 Maths Question 1.
sin²(2x+5)
Solution:
Ex 7.3 Class 12 Maths NCERT Solutions Q 1
Ex 7.3 Class 12 Maths Question 2.
sin3x cos4x
Solution:
Ex 7.3 Class 12 Maths NCERT Solutions Q 2
Ex 7.3 Class 12 Maths Question 3.
∫cos2x cos4x cos6x dx
Solution:
Ex 7.3 Class 12 Maths NCERT Solutions Q 3
Ex 7.3 Class 12 Maths Question 4.
∫sin3(2x+1)dx
Solution:
Ex 7.3 Class 12 Maths NCERT Solutions Q 4
Ex 7.3 Class 12 Maths Question 5.
sin3x cos3x
Solution:
NCERT Solutions Integration Chapter 7 Ex 7.3 Q 5
Ex 7.3 Class 12 Maths Question 6.
sinx sin2x sin3x
Solution:
NCERT Solutions Integration Chapter 7 Ex 7.3 Q 6
Ex 7.3 Class 12 Maths Question 7.
sin 4x sin 8x
Solution:
NCERT Solutions Integration Chapter 7 Ex 7.3 Q 7
Ex 7.3 Class 12 Maths Question 8.
[latex s=2]\frac { 1-cosx }{ 1+cosx }[/latex]
Solution:
chapter 7 maths class 12 NCERT Solutions Ex 7.3 Q 8
Ex 7.3 Class 12 Maths Question 9.
[latex s=2]\frac { cosx }{ 1+cosx }[/latex]
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 9
Ex 7.3 Class 12 Maths Question 10.
∫sinx4 dx
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 10
Ex 7.3 Class 12 Maths Question 11.
cos4 2x
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 11
Ex 7.3 Class 12 Maths Question 12.
[latex s=2]\frac { { sin }^{ 2 }x }{ 1+cosx } [/latex]
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 12
Ex 7.3 Class 12 Maths Question 13.
[latex s=2]\frac { cos2x-cos2\alpha }{ cosx-cos\alpha } [/latex]
Solution:
Class 12 Maths Chapter 7 Solutions Ex 7.3 Q 13
Ex 7.3 Class 12 Maths Question 14.
[latex s=2]\frac { cosx-sinx }{ 1+sin2x }[/latex]
Solution:
Chapter 7 Class 12 Maths NCERT Solutions Ex 7.3 Q 14
Ex 7.3 Class 12 Maths Question 15.
[latex s=2]\int { { tan }^{ 3 }2x\quad sec2x\quad dx=I } [/latex]
Solution:
Chapter 7 Class 12 Maths NCERT Solutions Ex 7.3 Q 15
Ex 7.3 Class 12 Maths Question 16.
tan4x
Solution:
NCERT Solutions for Class 12 Maths Ex 7.3 Q 16
Ex 7.3 Class 12 Maths Question 17.
[latex s=2]\frac { { sin }^{ 3 }x+{ cos }^{ 3 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Ex 7.3 Q 17
Ex 7.3 Class 12 Maths Question 18.
[latex s=2]\frac { cos2x+{ 2sin }^{ 2 }x }{ { cos }^{ 2 }x } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Ex 7.3 Q 18
Ex 7.3 Class 12 Maths Question 19.
[latex s=2]\frac { 1 }{ sinx{ cos }^{ 3 }x } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Ex 7.3 Q 19
Ex 7.3 Class 12 Maths Question 20.
[latex s=2]\frac { cos2x }{ { (cosx+sinx) }^{ 2 } } [/latex]
Solution:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.3 Q 20
Ex 7.3 Class 12 Maths Question 21.
sin-1 (cos x)
Solution:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.3 Q 21
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.3 Q 21 - i
Ex 7.3 Class 12 Maths Question 22.
[latex s=2]\int { \frac { 1 }{ cos(x-a)cos(x-b) } dx } [/latex]
Solution:
Class 12 Maths NCERT Solutions Chapter 7 Ex 7.3 Q 22
Ex 7.3 Class 12 Maths Question 23.
[latex s=2]\int { \frac { { sin }^{ 2 }x-{ cos }^{ 2 }x }{ { sin }^{ 2 }x{ cos }^{ 2 }x } } dx\quad is\quad equal\quad to[/latex]
(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
Ex 7.3 Class 12 Maths Question 24.
[latex s=2]\int { \frac { e^{ x }(1+x) }{ cos^{ 2 }({ e }^{ x }.{ x }) } } dx\quad is\quad equal\quad to[/latex]
(a) -cot(e.xx)+c
(b) tan(xex)+c
(c) tan(ex)+c
(d) cot ex+c
Solution:
NCERT Solution Class 12 Maths Ex 7.3 Q 24

Integrals Class 12 Ex 7.4

Ex 7.4 Class 12 Maths Question 1.
[latex s=2]\frac { { 3x }^{ 2 } }{ { x }^{ 6 }+1 } [/latex]
Solution:
Class 12 Integrals NCERT Solutions Ex 7.4 Q 1
Ex 7.4 Class 12 Maths Question 2.
[latex s=2]\frac { 1 }{ \sqrt { 1+{ 4x }^{ 2 } } } [/latex]
Solution:
Class 12 Integrals NCERT Solutions Ex 7.4 Q 2
Ex 7.4 Class 12 Maths Question 3.
[latex s=2]\frac { 1 }{ \sqrt { { (2-x) }^{ 2 }+1 } } [/latex]
Solution:
Class 12 Integrals NCERT Solutions Ex 7.4 Q 3
Ex 7.4 Class 12 Maths Question 4.
[latex s=2]\frac { 1 }{ \sqrt { 9-{ 25x }^{ 2 } } } [/latex]
Solution:
Class 12 Integrals NCERT Solutions Ex 7.4 Q 4
Ex 7.4 Class 12 Maths Question 5.
[latex s=2]\frac { 3x }{ 1+{ 2x }^{ 4 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 PDF Ex 7.4 Q 5
Ex 7.4 Class 12 Maths Question 6.
[latex s=2]\frac { { x }^{ 2 } }{ 1-{ x }^{ 6 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 PDF Ex 7.4 Q 6
Ex 7.4 Class 12 Maths Question 7.
[latex s=2]\frac { x-1 }{ \sqrt { { x }^{ 2 }-1 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 PDF Ex 7.4 Q 7
Ex 7.4 Class 12 Maths Question 8.
[latex s=2]\frac { { x }^{ 2 } }{ \sqrt { { x }^{ 6 }+{ a }^{ 6 } } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 PDF Ex 7.4 Q 8
Ex 7.4 Class 12 Maths Question 9.
[latex s=2]\frac { { sec }^{ 2 }x }{ \sqrt { { tan }^{ 2 }x+4 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 9
Ex 7.4 Class 12 Maths Question 10.
[latex s=2]\frac { 1 }{ \sqrt { { x }^{ 2 }+2x+2 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 10
Ex 7.4 Class 12 Maths Question 11.

[latex s=2]\frac { 1 }{ { 9x }^{ 2 }+6x+5 } [/latex]
Solution:
miscellaneous exercise on chapter 7 class 12 Q 11
Ex 7.4 Class 12 Maths Question 12.
[latex s=2]\frac { 1 }{ \sqrt { 7-6x-{ x }^{ 2 } } } [/latex]
Solution:
miscellaneous exercise on chapter 7 class 12 Q 12
Ex 7.4 Class 12 Maths Question 13.
[latex s=2]\frac { 1 }{ \sqrt { (x-1)(x-2) } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 13
Ex 7.4 Class 12 Maths Question 14.
[latex s=2]\frac { 1 }{ \sqrt { 8+3x-{ x }^{ 2 } } } [/latex]
Solution:
miscellaneous exercise on chapter 7 class 12 Q 14
Ex 7.4 Class 12 Maths Question 15.
[latex s=2]\frac { 1 }{ \sqrt { (x-a)(x-b) } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Integrals Ex 7.4 Q 15
Ex 7.4 Class 12 Maths Question 16.
[latex s=2]\frac { 4x+1 }{ \sqrt { { 2x }^{ 2 }+x-3 } } [/latex]
Solution:
Class 12 Maths Solutions Ex 7.4 Q 16
Ex 7.4 Class 12 Maths Question 17.
[latex s=2]\frac { x+2 }{ \sqrt { { x }^{ 2 }-1 } } [/latex]
Solution:
Class 12 Maths Solutions Ex 7.4 Q 17
Class 12 Maths Solutions Ex 7.4 Q 17 - i
Ex 7.4 Class 12 Maths Question 18.
[latex s=2]\frac { 5x-2 }{ 1+2x+{ 3x }^{ 2 } } [/latex]
Solution:
put 5x-2=A[latex s=2]\frac { d }{ dx }[/latex](1+2x+3x²)+B
⇒ 6A=5, A=[latex s=2]\frac { 5 }{ 6 }-2=2A+B[/latex], B=[latex s=2]-\frac { 11 }{ 3 }[/latex]
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 18

Ex 7.4 Class 12 Maths Question 19.
[latex s=2]\frac { 6x+7 }{ \sqrt { (x-5)(x-4) } } [/latex]
Solution:
Class 12 Maths Solutions Ex 7.4 Q 18
Class 12 Maths Solutions Ex 7.4 Q 18 - i
Class 12 Maths Solutions Ex 7.4 Q 18 - ii
Ex 7.4 Class 12 Maths Question 20.
[latex s=2]\frac { x+2 }{ \sqrt { 4x-{ x }^{ 2 } } } [/latex]
Solution:
Class 12 Maths Solutions Ex 7.4 Q 19
Class 12 Maths Solutions Ex 7.4 Q 19 - i
Class 12 Maths Solutions Ex 7.4 Q 19 - ii
Ex 7.4 Class 12 Maths Question 21.
[latex s=2]\frac { x+2 }{ \sqrt { { x }^{ 2 }+2x+3 } } [/latex]
Solution:
Class 12 Maths Solutions Ex 7.4 Q 20
Class 12 Maths Solutions Ex 7.4 Q 20 - i
Ex 7.4 Class 12 Maths Question 22.
[latex s=2]\frac { x+3 }{ { x }^{ 2 }-2x-5 } [/latex]
Solution:
NCERT Maths Class 12 Ex 7.4 Q 21
NCERT Maths Class 12 Ex 7.4 Q 21 -i
Ex 7.4 Class 12 Maths Question 23.
[latex s=2]\frac { 5x+3 }{ \sqrt { { x }^{ 2 }+4x+10 } } [/latex]
Solution:
NCERT Maths Class 12 Ex 7.4 Q 22
NCERT Maths Class 12 Ex 7.4 Q 22 - i
Ex 7.4 Class 12 Maths Question 24.
[latex s=2]\int { \frac { dx }{ { x }^{ 2 }+2x+2 } equals } [/latex]
(a) xtan-1(x+1)+c
(b) (x+1)tan-1x+c
(c) tan-1(x+1)+c
(d) tan-1x+c
Solution:
NCERT Maths Class 12 Ex 7.4 Q 23
Ex 7.4 Class 12 Maths Question 25.
[latex s=2]\int { \frac { dx }{ \sqrt { 9x-{ 4x }^{ 2 } } } equals } [/latex]
(a) [latex s=2]\frac { 1 }{ 9 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c[/latex]
(b) [latex s=2]\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 8x-9 }{ 9 } \right) +c[/latex]
(c) [latex s=2]\frac { 1 }{ 3 } { sin }^{ -1 }\left( \frac { 9x-8 }{ 8 } \right) +c[/latex]
(d) [latex s=2]{ sin }^{ -1 }\left( \frac { 9x-8 }{ 9 } \right) +c[/latex]
Solution:

Integrals Class 12 Ex 7.5

Ex 7.5 Class 12 Maths Question 1.
[latex s=2]\frac { x }{ (x+1)(x+2) }[/latex]
Solution:
NCERT Solutions Maths Class 12 Ex 7.5 Q 1
Ex 7.5 Class 12 Maths Question 2.
[latex s=2]\frac { 1 }{ { x }^{ 2 }-9 } [/latex]
Solution:
NCERT Solutions Maths Class 12 Ex 7.5 Q 2
Ex 7.5 Class 12 Maths Question 3.
[latex s=2]\frac { 3x-1 }{ (x-1)(x-2)(x-3) }[/latex]
Solution:
NCERT Solutions Maths Class 12 Ex 7.5 Q 3
Ex 7.5 Class 12 Maths Question 4.
[latex s=2]\frac { x }{ (x-1)(x-2)(x-3) }[/latex]
Solution:
NCERT Solutions Maths Class 12 Ex 7.5 Q 4
Ex 7.5 Class 12 Maths Question 5.
[latex s=2]\frac { 2x }{ { x }^{ 2 }+3x+2 } [/latex]
Solution:
Chapter 7 Class 12 Maths Ex 7.5 Q 5
Ex 7.5 Class 12 Maths Question 6.
[latex s=2]\frac { 1-{ x }^{ 2 } }{ x(1-2x) } [/latex]
Solution:
Chapter 7 Class 12 Maths Ex 7.5 Q 6
Ex 7.5 Class 12 Maths Question 7.
[latex s=2]\frac { x }{ \left( { x }^{ 2 }+1 \right) \left( x-1 \right) } [/latex]
Solution:
Chapter 7 Class 12 Maths Ex 7.5 Q 7
Chapter 7 Class 12 Maths Ex 7.5 Q 7 - i
Ex 7.5 Class 12 Maths Question 8.
[latex s=2]\frac { x }{ { \left( x-1 \right) }^{ 2 }\left( x+2 \right) } [/latex]
Solution:
Class 12 Integration NCERT Solutions Ex 7.5 Q 8
Ex 7.5 Class 12 Maths Question 9.
[latex s=2]\frac { 3x+5 }{ { x }^{ 3 }-{ x }^{ 2 }-x+1 } [/latex]
Solution:
Class 12 Integration NCERT Solutions Ex 7.5 Q 9
Ex 7.5 Class 12 Maths Question 10.
[latex s=2]\frac { 2x-3 }{ ({ x }^{ 2 }-1)(2x+3) } [/latex]
Solution:
Class 12 Integration NCERT Solutions Ex 7.5 Q 10
Ex 7.5 Class 12 Maths Question 11.
[latex s=2]\frac { 5x }{ (x-1)({ x }^{ 2 }-4) } [/latex]
Solution:
Class 12 Integration NCERT Solutions Ex 7.5 Q 11
Ex 7.5 Class 12 Maths Question 12.
[latex s=2]\frac { { x }^{ 3 }+x+1 }{ { x }^{ 2 }-1 } [/latex]
Solution:
Class 12 Integration NCERT Solutions Ex 7.5 Q 12
Ex 7.5 Class 12 Maths Question 13.
[latex s=2]\frac { 2 }{ (1-x)(1+{ x }^{ 2 }) } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Q 13
Ex 7.5 Class 12 Maths Question 14.
[latex s=2]\frac { 3x-1 }{ { (x+2) }^{ 2 } } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Q 14
Ex 7.5 Class 12 Maths Question 15.
[latex s=2]\frac { 1 }{ { x }^{ 4 }-1 } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Miscellaneous Exercise Q 15
Ex 7.5 Class 12 Maths Question 16.
[latex s=2]\frac { 1 }{ x({ x }^{ n }+1) } [/latex]
[Hint : multiply numerator and denominator by xn-1 and put xn = t ]
Solution:
Class 12 Maths Ex 7.5 Q 16
Class 12 Maths Ex 7.5 Q 16 - i
Ex 7.5 Class 12 Maths Question 17.
[latex s=2]\frac { cosx }{ (1-sinx)(2-sinx) } [/latex]
Solution:
Class 12 Maths Ex 7.5 Q 17
Ex 7.5 Class 12 Maths Question 18.
[latex s=2]\frac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right) }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right) } [/latex]
Solution:
Class 12 Maths Ex 7.5 Q 18
Class 12 Maths Ex 7.5 Q 18 - i
Ex 7.5 Class 12 Maths Question 19.
[latex s=2]\frac { 2x }{ ({ x }^{ 2 }+1)({ x }^{ 2 }+3) } [/latex]
Solution:
NCERT Class 12 Maths Ex 7.5 Q 19
Ex 7.5 Class 12 Maths Question 20.
[latex s=2]\frac { 1 }{ x({ x }^{ 4 }-1) } [/latex]
Solution:
NCERT Class 12 Maths Ex 7.5 Q 20
Ex 7.5 Class 12 Maths Question 21.
[latex s=2]\frac { 1 }{ { e }^{ x }-1 } [/latex]
Solution:
NCERT Class 12 Maths Ex 7.5 Q 21
Ex 7.5 Class 12 Maths Question 22.
choose the correct answer in each of the following :
[latex s=2]\int { \frac { xdx }{ (x-1)(x-2) } equals } [/latex]
(a) [latex s=2]log\left| \frac { { (x-1) }^{ 2 } }{ x-2 } \right| +c[/latex]
(b) [latex s=2]log\left| \frac { { (x-2) }^{ 2 } }{ x-1 } \right| +c[/latex]
(c) [latex s=2]log\left| \left( \frac { x-{ 1 }^{ 2 } }{ x-2 } \right) \right| +c[/latex]
(d) log|(x-1)(x-2)|+c
Solution:
NCERT Class 12 Maths Ex 7.5 Q 22
Ex 7.5 Class 12 Maths Question 23.
[latex s=2]\int { \frac { dx }{ x({ x }^{ 2 }+1) } equals } [/latex]
(a) [latex s=2]log|x|-\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c [/latex]
(b) [latex s=2]log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c [/latex]
(c) [latex s=2]-log|x|+\frac { 1 }{ 2 } log({ x }^{ 2 }+1)+c[/latex]
(d) [latex s=2]\frac { 1 }{ 2 } log|x|+log({ x }^{ 2 }+1)+c [/latex]
Solution:
NCERT Class 12 Maths Ex 7.5 Q 23


Integrals Class 12 Ex 7.6

Ex 7.6 Class 12 Maths Question 1.
x sinx
Solution:
Maths Class 12 NCERT Solutions Chapter 7 Ex 7.6 Q 1
Ex 7.6 Class 12 Maths Question 2.
x sin3x
Solution:
Maths Class 12 NCERT Solutions Chapter 7 Ex 7.6 Q 2
Ex 7.6 Class 12 Maths Question 3.
[latex s=2]{ x }^{ 2 }{ e }^{ x }[/latex]
Solution:
Maths Class 12 NCERT Solutions Chapter 7 Ex 7.6 Q 3
Ex 7.6 Class 12 Maths Question 4.
x logx
Solution:
Maths Class 12 NCERT Solutions Chapter 7 Ex 7.6 Q 4
Ex 7.6 Class 12 Maths Question 5.
x log2x
Solution:
NCERT Maths Solution Class 12 Ex 7.6 Q 5
Ex 7.6 Class 12 Maths Question 6.
[latex s=2]{ x }^{ 2 }logx[/latex]
Solution:
NCERT Maths Solution Class 12 Ex 7.6 Q 6
Ex 7.6 Class 12 Maths Question 7.
[latex s=2]x\quad { sin }^{ -1 }x[/latex]
Solution:
NCERT Maths Solution Class 12 Ex 7.6 Q 7
Ex 7.6 Class 12 Maths Question 8.
[latex s=2]x\quad { tan }^{ -1 }x[/latex]
Solution:
NCERT Maths Solution Class 12 Ex 7.6 Q 8
Ex 7.6 Class 12 Maths Question 9.
[latex s=2]x\quad { cos }^{ -1 }x[/latex]
Solution:
Class 12 Maths NCERT Solution Ex 7.6 Q 9
Class 12 Maths NCERT Solution Ex 7.6 Q 9 - i
Ex 7.6 Class 12 Maths Question 10.
[latex s=2]{ (sin }^{ -1 }{ x })^{ 2 }[/latex]
Solution:
Class 12 Maths NCERT Solution Ex 7.6 Q 10
Ex 7.6 Class 12 Maths Question 11.
[latex s=2]\frac { x\quad { cos }^{ -1 }x }{ \sqrt { 1-{ x }^{ 2 } } } [/latex]
Solution:
Class 12 Maths NCERT Solution Ex 7.6 Q 11
Ex 7.6 Class 12 Maths Question 12.
x sec²x
Solution:
Class 12 Maths NCERT Solution Ex 7.6 Q 12
Ex 7.6 Class 12 Maths Question 13.
[latex s=2]{ ta }n^{ -1 }x[/latex]
Solution:
Class 12 Maths NCERT Solution Q 13
Ex 7.6 Class 12 Maths Question 14.
x(logx)²
Solution:
Class 12 Maths NCERT Solution Q 14
Ex 7.6 Class 12 Maths Question 15.
(x²+1)logx
Solution:
Class 12 Maths NCERT Solution Q 15
Class 12 Maths NCERT Solution Q 15 - i
Ex 7.6 Class 12 Maths Question 16.
[latex s=2]{ e }^{ x }(sinx+cosx)[/latex]
Solution:
Class 12 Maths NCERT Solution Q 16
Ex 7.6 Class 12 Maths Question 17.
[latex s=2]\frac { { xe }^{ x } }{ { (1+x) }^{ 2 } } [/latex]
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 17
Ex 7.6 Class 12 Maths Question 18.
[latex s=2]\frac { { e }^{ x }(1+sinx) }{ 1+cosx } [/latex]
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 18
Ch 7 Maths Class 12 Ex 7.6 Q 18 - i
Ex 7.6 Class 12 Maths Question 19.
[latex s=2]{ e }^{ x }\left( \frac { 1 }{ x } -\frac { 1 }{ { x }^{ 2 } } \right) [/latex]
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 19
Ex 7.6 Class 12 Maths Question 20.
[latex s=2]\frac { { (x-2)e }^{ x } }{ { (x-1) }^{ 3 } } [/latex]
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 20
Ex 7.6 Class 12 Maths Question 21.
[latex s=2]{ e }^{ 2x }sinx[/latex]
Solution:
Ch 7 Maths Class 12 Ex 7.6 Q 21
Ex 7.6 Class 12 Maths Question 22.
[latex s=2]{ sin }^{ -1 }\left( \frac { 2x }{ 1+{ x }^{ 2 } } \right) [/latex]
Solution:
Maths Class 12 Chapter 7 NCERT Solutions Ex 7.6 Q 22

Choose the correct answer in exercise 23 and 24

Ex 7.6 Class 12 Maths Question 23.
[latex s=2]\int { { x }^{ 2 }{ e }^{ { x }^{ 3 } } } dx\quad equals[/latex]
(a) [latex s=2]\frac { 1 }{ 3 } { e }^{ { x }^{ 3 } }+c[/latex]
(b) [latex s=2]\frac { 1 }{ 3 } +{ e }^{ { x }^{ 2 } }+c[/latex]
(c) [latex s=2]\frac { 1 }{ 2 } { e }^{ { x }^{ 3 } }+c[/latex]
(d) [latex s=2]\frac { 1 }{ 2 } { e }^{ { x }^{ 2 } }+c[/latex]
Solution:
Maths Class 12 Chapter 7 NCERT Solutions Ex 7.6 Q 23
Ex 7.6 Class 12 Maths Question 24.
[latex s=2]\int { { e }^{ x }secx(1+tanx) } dx\quad equals[/latex]
(a) [latex s=2]{ e }^{ x }cosx+c[/latex]
(b) [latex s=2]{ e }^{ x }secx+c[/latex]
(c) [latex s=2]{ e }^{ x }sinx+c[/latex]
(d) [latex s=2]{ e }^{ x }tanx+c[/latex]
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

Ex 7.7 Class 12 Maths Question 1.
[latex s=2]\sqrt { 4-{ x }^{ 2 } } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.7 Q 1
Ex 7.7 Class 12 Maths Question 2.
[latex s=2]\sqrt { 1-{ 4x }^{ 2 } } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.7 Q 2
Ex 7.7 Class 12 Maths Question 3.
[latex s=2]\sqrt { { x }^{ 2 }+4x+6 } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.7 Q 3
Ex 7.7 Class 12 Maths Question 4.
[latex s=2]\sqrt { { x }^{ 2 }+4x+1 } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.7 Q 4
Ex 7.7 Class 12 Maths Question 5.
[latex s=2]\sqrt { 1-4x-{ x }^{ 2 } } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.7 Q 5
Ex 7.7 Class 12 Maths Question 6.
[latex s=2]\sqrt { { x }^{ 2 }+4x-5 } [/latex]
Solution:
NCERT Solutions Class 12 Ex 7.7 Q 6
Ex 7.7 Class 12 Maths Question 7.
[latex s=2]\sqrt { 1+3x-{ x }^{ 2 } } [/latex]
Solution:
NCERT Solutions Class 12 Ex 7.7 Q 7
Ex 7.7 Class 12 Maths Question 8.
[latex s=2]\sqrt { { x }^{ 2 }+3x } [/latex]
Solution:
NCERT Solutions Class 12 Ex 7.7 Q 8
Ex 7.7 Class 12 Maths Question 9.
[latex s=2]\sqrt { 1+\frac { { x }^{ 2 } }{ 9 } } [/latex]
Solution:
NCERT Solutions Class 12 Ex 7.7 Q 9
Choose the correct answer in the Exercises 10 to 11:

Ex 7.7 Class 12 Maths Question 10.
[latex s=2]\int { \sqrt { 1+{ x }^{ 2 } } } dx\quad is\quad equal\quad to[/latex]
(a) [latex s=2]\frac { x }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } log|x+\sqrt { 1+{ x }^{ 2 } } |+c[/latex]
(b) [latex s=2]\frac { 2 }{ 3 } { \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c[/latex]
(c) [latex s=2]\frac { 2 }{ 3 } x{ \left( 1+{ x }^{ 2 } \right) }^{ \frac { 3 }{ 2 } }+c[/latex]
(d) [latex s=2]\frac { { x }^{ 2 } }{ 2 } \sqrt { 1+{ x }^{ 2 } } +\frac { 1 }{ 2 } { x }^{ 2 }log\left| x+\sqrt { 1+{ x }^{ 2 } } \right| +c[/latex]
Solution:
Exercise 7.7 Class 12 Maths Q 10
Ex 7.7 Class 12 Maths Question 11.
[latex s=2]\int { \sqrt { { x }^{ 2 }-8x+7 } } dx\quad is\quad equal\quad to[/latex]
NCERT Solutions for Class 12 Maths Chapter 7 Integrals 11
Solution:
Exercise 7.7 Class 12 Maths Q 11


Class 12 Integrals Ex 7.8

Ex 7.8 Class 12 Maths Question 1.
[latex s=2]\int _{ a }^{ b }{ x\quad dx } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.8 Q 1
Maths NCERT Class 12 Ex 7.8 Q 1 - i

Ex 7.8 Class 12 Maths Question 2.
[latex s=2]\int _{ 0 }^{ 5 }{ (x+1)dx } [/latex]
Solution:
Maths NCERT Class 12 Ex 7.8 Q 2
Maths NCERT Class 12 Ex 7.8 Q 2 - i

Ex 7.8 Class 12 Maths Question 3.
[latex s=2]\int _{ 2 }^{ 3 }{ { x }^{ 2 } } dx[/latex]
Solution:
Maths NCERT Class 12 Ex 7.8 Q 3
Maths NCERT Class 12 Ex 7.8 Q 3 -i

Ex 7.8 Class 12 Maths Question 4.
[latex s=2]\int _{ 1 }^{ 4 }{ ({ x }^{ 2 }-x) } dx[/latex]
Solution:
NCERT Solutions Class 12 Ex 7.8 Q 4
NCERT Solutions Class 12 Ex 7.8 Q 4 - i
NCERT Solutions Class 12 Ex 7.8 Q 4 - ii

Ex 7.8 Class 12 Maths Question 5.
[latex s=2]\int _{ -1 }^{ 1 }{ { e }^{ x } } dx\quad [/latex]
Solution:
Maths NCERT Class 12 Ex 7.8 Q 5
Maths NCERT Class 12 Ex 7.8 Q 5 - i

Ex 7.8 Class 12 Maths Question 6.
[latex s=2]\int _{ 0 }^{ 4 }{ { (x+e }^{ 2x }) } dx\quad [/latex]
Solution:
Maths NCERT Class 12 Ex 7.8 Q 6
Maths NCERT Class 12 Ex 7.8 Q 6 - i


Class 12 Integrals Ex 7.9

Ex 7.9 Class 12 Maths Question 1.
[latex s=2]\int _{ -1 }^{ 1 }{ { (x+1 }) } dx\quad [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 1

Ex 7.9 Class 12 Maths Question 2.
[latex s=2]\int _{ 2 }^{ 3 }{ \frac { 1 }{ x } dx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 2

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

Ex 7.9 Class 12 Maths Question 4.
[latex s=2]\int _{ 0 }^{ \frac { \pi }{ 4 } }{ sin2x\quad dx } [/latex]
Solution:
NCERT Solutions for Class 12 Maths Chapter 7 Ex 7.9 Q 4

Ex 7.9 Class 12 Maths Question 5.
[latex s=2]\int _{ 0 }^{ \frac { \pi }{ 2 } }{ cos2x\quad dx } [/latex]
Solution:
NCERT Integration Ex 7.9 Q 5

Ex 7.9 Class 12 Maths Question 6.
[latex s=2]\int _{ 4 }^{ 5 }{ { e }^{ x }dx } [/latex]
Solution:
NCERT Integration Ex 7.9 Q 6

Ex 7.9 Class 12 Maths Question 7.
[latex s=2]\int _{ 0 }^{ \frac { \pi }{ 4 } }{ tanx\quad dx } [/latex]
Solution:
NCERT Integration Ex 7.9 Q 7

Ex 7.9 Class 12 Maths Question 8.
[latex s=2]\int _{ \frac { \pi }{ 6 } }^{ \frac { \pi }{ 4 } }{ cosec\quad xdx } [/latex]
Solution:
NCERT Integration Ex 7.9 Q 8

Ex 7.9 Class 12 Maths Question 9.
[latex s=2]\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 1-{ x }^{ 2 } } } } [/latex]
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 9

Ex 7.9 Class 12 Maths Question 10.
[latex s=2]\int _{ 0 }^{ 1 }{ \frac { dx }{ 1+{ x }^{ 2 } } } [/latex]
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 10

Ex 7.9 Class 12 Maths Question 11.
[latex s=2]\int _{ 2 }^{ 3 }{ \frac { dx }{ { x }^{ 2 }-1 } } [/latex]
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 11

Ex 7.9 Class 12 Maths Question 12.
[latex s=2]\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { cos }^{ 2 } } xdx[/latex]
Solution:
NCERT Class 12 Maths Solution Ex 7.9 Q 12

Ex 7.9 Class 12 Maths Question 13.
[latex s=2]\int _{ 2 }^{ 3 }{ \frac { x }{ { x }^{ 2 }+1 } } dx[/latex]
Solution:
Ex 7.9 Class 12 Q 13

Ex 7.9 Class 12 Maths Question 14.
[latex s=2]\int _{ 0 }^{ 1 }{ \frac { 2x+3 }{ { 5x }^{ 2 }+1 } dx } [/latex]
Solution:
Ex 7.9 Class 12 Q 14

Ex 7.9 Class 12 Maths Question 15.
[latex s=2]\int _{ 0 }^{ 1 }{ { xe }^{ { x }^{ 2 } }dx } [/latex]
Solution:
Ex 7.9 Class 12 Q 15

Ex 7.9 Class 12 Maths Question 16.
[latex s=2]\int _{ 1 }^{ 2 }{ \frac { { 5x }^{ 2 } }{ { x }^{ 2 }+4x+3 } dx } [/latex]
Solution:
Ex 7.9 Class 12 Q 16
Ex 7.9 Class 12 Q 16 - i

Ex 7.9 Class 12 Maths Question 17.
[latex s=2]\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \left( { 2sec }^{ 2 }x+{ x }^{ 3 }+2 \right) dx } [/latex]
Solution:
Ex 7.9 Class 12 Q 17

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

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

Ex 7.9 Class 12 Maths Question 20.
[latex s=2]\int _{ 0 }^{ 1 }{ \left( { xe }^{ x }+sin\frac { \pi x }{ 4 } \right) dx } [/latex]
Solution:
Maths NCERT Class 12 Solutions Ex 7.9 Q 20

Ex 7.9 Class 12 Maths Question 21.
[latex s=2]\int _{ 1 }^{ \sqrt { 3 } }{ \frac { dx }{ { 1+x }^{ 2 } } \quad equals } [/latex]
(a) [latex s=2]\frac { \pi }{ 3 } [/latex]
(b) [latex s=2]\frac { 2\pi }{ 3 } [/latex]
(c) [latex s=2]\frac { \pi }{ 6 } [/latex]
(d) [latex s=2]\frac { \pi }{ 12 } [/latex]
Solution:
Maths NCERT Class 12 Solutions Ex 7.9 Q 21

Ex 7.9 Class 12 Maths Question 22.
[latex s=2]\int _{ 0 }^{ \frac { 2 }{ 3 } }{ \frac { dx }{ 4+{ 9x }^{ 2 } } equals } [/latex]
(a) [latex s=2]\frac { \pi }{ 6 }[/latex]
(b) [latex s=2]\frac { \pi }{ 12 }[/latex]
(c) [latex s=2]\frac { \pi }{ 24 }[/latex]
(d) [latex s=2]\frac { \pi }{ 4 }[/latex]
Solution:
Maths NCERT Class 12 Solutions Ex 7.9 Q 22


Integration Class 12 Ex 7.10

Ex 7.10 Class 12 Maths 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
Ex 7.10 Class 12 Maths Question 2:
Ex 7.10 Integrals Class 12 NCERT Solutions Ex 7.10 Q 2
Solution:
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 2
Ex 7.10 Class 12 Maths Question 3:
Ex 7.10 Integrals Class 12 NCERT Solutions Ex 7.10 Q 3
Solution:
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 3
Ex 7.10 Class 12 Maths Question 4:
Integration Class 12 Ex 7.10 Q 4
Solution:
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 4
Ex 7.10 Class 12 Maths Question 5:
Integration Class 12 Ex 7.10 Q 5
Solution:
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 5
NCERT Solutions of Class 12 Maths Chapter 7 Integrals Ex 7.10 Q 5 - i
Ex 7.10 Class 12 Maths Question 6:
Integration Class 12 Ex 7.10 Q 6
Solution:
NCERT Class 12 Solutions Integrals Ex 7.10 Q 6
NCERT Class 12 Solutions Integrals Ex 7.10 Q 6 - i
Ex 7.10 Class 12 Maths Question 7:
Integration Class 12 Ex 7.10 Q 7
Solution:
NCERT Class 12 Solutions Integrals Ex 7.10 Q 7
Ex 7.10 Class 12 Maths Question 8:
Integrals Class 12 Ex 7.10 8
Solution:
NCERT Class 12 Solutions Integrals Ex 7.10 Q 8
Ex 7.10 Class 12 Maths Question 9:
Integration Class 12 Ex 7.10 Q 9
Solution:
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 9
Ex 7.10 Class 12 Maths Question 10:
Integrals Class 12 Ex 7.10 10
Solution:
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 10
Ex 7.10 Class 12 Maths Question 11:
Integrals Class 12 Ex 7.10 11
Solution:
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 11
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 11 - i
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 11 - ii
ncert solution of maths class 12 Chapter 7 Ex 7.10 Q 11 - iii
Ex 7.10 Class 12 Maths Question 12:
integrals class 12 ncert solutions Ex 7.10 Q 12
Solution:
NCERT Class 12 Solutions Ex 7.10 Q 12
NCERT Class 12 Solutions Ex 7.10 Q 12 - i
NCERT Class 12 Solutions Ex 7.10 Q 12 - ii
Ex 7.10 Class 12 Maths Question 13:
integrals class 12 ncert solutions Ex 7.10 Q 13
Solution:
NCERT Class 12 Solutions Ex 7.10 Q 13
Ex 7.10 Class 12 Maths Question 14:
integrals class 12 ncert solutions Ex 7.10 Q 14
Solution:
NCERT Class 12 Solutions Ex 7.10 Q 14
Ex 7.10 Class 12 Maths Question 15:
integrals class 12 ncert solutions Ex 7.10 Q 15
Solution:
NCERT Class 12 Solutions Ex 7.10 Q 15
Ex 7.10 Class 12 Maths Question 16:
maths ncert solutions class 12 Ex 7.10 Q 16
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 16
Ex 7.10 Class 12 Maths Question 17:
maths ncert solutions class 12 Ex 7.10 Q 17
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 17
NCERT Maths 12 Solutions Ex 7.10 Q 17 - i
Ex 7.10 Class 12 Maths Question 18:
maths ncert solutions class 12 Ex 7.10 Q 18
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 18
NCERT Maths 12 Solutions Ex 7.10 Q 18 - i
Ex 7.10 Class 12 Maths Question 19:
maths ncert solutions class 12 Ex 7.10 Q 19
maths ncert solutions class 12 Ex 7.10 Q 19 - i
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 19
Ex 7.10 Class 12 Maths Question 20:
maths ncert solutions class 12 Ex 7.10 Q 20
maths ncert solutions class 12 Ex 7.10 Q 20 - i
Solution:
NCERT Maths 12 Solutions Ex 7.10 Q 20


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
class 12 maths ncert solutions Ex 7.11 Q 2
Ex 7.11 Integrals Class 12 Maths NCERT Solutions Sol 2
class 12 maths ncert solutions Ex 7.11 Q 3
Ex 7.11 Integrals Class 12 Maths NCERT Solutions Sol 3
class 12 maths ncert solutions Ex 7.11 Q 4
Ex 7.11 Integrals Class 12 Maths NCERT Solutions Sol 4
integrals ncert solutions Ex 7.11 Q 5
Ex 7.11 Integrals Class 12 Maths NCERT Solutions Sol 5
integrals ncert solutions Ex 7.11 Q 6
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 6
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 6
integrals ncert solutions Ex 7.11 Q 7
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 7
integrals ncert solutions Ex 7.11 Q 8
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 8
integrals ncert solutions Ex 7.11 Q 9
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 9
integrals ncert solutions Ex 7.11 Q 10
Ex 7.11 Integration Class 12 Maths NCERT Solutions Sol 10
integrals ncert solutions Ex 7.11 Q 11
NCERT maths solutions class 12 Ex 7.11 Sol 11
integration class 12 ncert solutions Ex 7.11 Q 12
NCERT maths solutions class 12 Ex 7.11 Sol 12
integration class 12 ncert solutions Ex 7.11 Q 13
NCERT maths solutions class 12 Ex 7.11 Sol 13
integration class 12 ncert solutions Ex 7.11 Q 14
NCERT maths solutions class 12 Ex 7.11 Sol 14
integration class 12 ncert solutions Ex 7.11 Q 15
NCERT maths solutions class 12 Ex 7.11 Sol 15
integration class 12 ncert solutions Ex 7.11 Q 16
NCERT solutions for class 12 maths Ex 7.11 Sol 16
integration class 12 ncert solutions Ex 7.11 Q 17
NCERT solutions for class 12 maths Ex 7.11 Sol 17
integration class 12 ncert solutions Ex 7.11 Q 18
NCERT solutions for class 12 maths Ex 7.11 Sol 18
integration class 12 ncert solutions Ex 7.11 Q 19
NCERT solutions for class 12 maths Ex 7.11 Sol 19
integration class 12 ncert solutions Ex 7.11 Q 20
NCERT solutions for class 12 maths Ex 7.11 Sol 20
integration class 12 ncert solutions Ex 7.11 Q 21
NCERT solutions for class 12 maths Ex 7.11 Sol 21
integration class 12 ncert solutions Ex 7.11 Q 22
NCERT solutions for class 12 maths Ex 7.11 Sol 22
integration class 12 ncert solutions Ex 7.11 Q 23
NCERT solutions for class 12 maths Ex 7.11 Sol 23
integration class 12 ncert solutions Ex 7.11 Q 24
NCERT solutions for class 12 maths Ex 7.11 Sol 24
integration class 12 ncert solutions Ex 7.11 Q 25
NCERT solutions for class 12 maths Ex 7.11 Sol 25
integration class 12 ncert solutions Ex 7.11 Q 26
Maths NCERT Solution Class 12 Ex 7.11 Q 26
integration class 12 ncert solutions Ex 7.11 Q 27
Maths NCERT Solution Class 12 Ex 7.11 Q 27
integration class 12 ncert solutions Ex 7.11 Q 28
Maths NCERT Solution Class 12 Ex 7.11 Q 28
integration class 12 ncert solutions Ex 7.11 Q 29
Maths NCERT Solution Class 12 Ex 7.11 Q 29
integration class 12 ncert solutions Ex 7.11 Q 30
Maths NCERT Solution Class 12 Ex 7.11 Q 30
Maths NCERT Solution Class 12 Ex 7.11 Q 30 - i
integration class 12 ncert solutions Ex 7.11 Q 31
Maths NCERT Solution Class 12 Ex 7.11 Q 31
Maths NCERT Solution Class 12 Ex 7.11 Q 31 - i
integration class 12 ncert solutions Ex 7.11 Q 32
Maths NCERT Solution Class 12 Ex 7.11 Q 32
integration class 12 ncert solutions Ex 7.11 Q 33
Maths NCERT Solution Class 12 Ex 7.11 Q 33
integration class 12 ncert solutions Ex 7.11 Q 34
Maths NCERT Solution Class 12 Ex 7.11 Q 34
integration class 12 ncert solutions Ex 7.11 Q 35
Maths NCERT Solution Class 12 Ex 7.11 Q 35
integration class 12 ncert solutions Ex 7.11 Q 36
Maths NCERT Solution Class 12 Ex 7.11 Q 36
integration class 12 ncert solutions Ex 7.11 Q 37
Maths NCERT Solution Class 12 Ex 7.11 Q 37
integration class 12 ncert solutions Ex 7.11 Q 38
integration class 12 ncert solutions Ex 7.11 Q 38 - i
Maths NCERT Solution Class 12 Ex 7.11 Q 38
integration class 12 ncert solutions Ex 7.11 Q 39
Maths NCERT Solution Class 12 Ex 7.11 Q 39
integration class 12 ncert solutions Ex 7.11 Q 40
integration class 12 ncert solutions Ex 7.11 Q 40 - i
Maths NCERT Solution Class 12 Ex 7.11 Q 40
integration class 12 ncert solutions Ex 7.11 Q 41
integration class 12 ncert solutions Ex 7.11 Q 41- i
Maths NCERT Solution Class 12 Ex 7.11 Q 41


NCERT Solutions for Class 12 Maths Chapter 7 Integrals Miscellaneous Exercise

Miscellaneous Exercise Class 11 Maths Question 1:
ncert solutions class 12 maths Miscellaneous Questions Q 1
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 1 - i
ncert solutions class 12 maths Miscellaneous Questions Q 1 - ii

Miscellaneous Exercise Class 11 Maths Question 2:
ncert solutions class 12 maths Miscellaneous Questions Q 2
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 2 - i

Miscellaneous Exercise Class 11 Maths Question 3:
miscellaneous exercise on chapter 7 class 12 Q 3
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 3 - i

vMiscellaneous Exercise Class 11 Maths Question 4:
ncert solutions class 12 maths Miscellaneous Questions Q 4
Solution:
miscellaneous exercise on chapter 7 class 12 Q 4 - i
ncert solutions class 12 maths Miscellaneous Questions Q 4 - ii

Miscellaneous Exercise Class 11 Maths Question 5:
ncert solutions class 12 maths Miscellaneous Questions Q 5
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 5 - i

Miscellaneous Exercise Class 11 Maths Question 6:
ncert solutions class 12 maths Miscellaneous Questions Q 6
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 6 - i

Miscellaneous Exercise Class 11 Maths Question 7:
ncert solutions class 12 maths Miscellaneous Question Q 7
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 7 - i

Miscellaneous Exercise Class 11 Maths Question 8:
ncert solutions class 12 maths Miscellaneous Questions Q 8
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 8 - i

Miscellaneous Exercise Class 11 Maths Question 9:
ncert solutions for class 12 maths chapter 7 miscellaneous exercise Q 9
Solution:
ncert solutions for class 12 maths miscellaneous Q 9 - i

Miscellaneous Exercise Class 11 Maths Question 10:
ncert solutions class 12 maths Miscellaneous Questions Q 10
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 10 - i
Miscellaneous Exercise Class 11 Maths Question 11:
ncert solutions class 12 maths Miscellaneous Questions Q 11
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 11 - i

Miscellaneous Exercise Class 11 Maths Question 12:
ncert solutions class 12 maths Miscellaneous Questions Q 12
Solution:
ncert miscellaneous solutions class 12 Q 12 - i

Miscellaneous Exercise Class 11 Maths Question 13:
ncert solutions class 12 maths Miscellaneous Questions Q 13
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 13 - i

Miscellaneous Exercise Class 11 Maths Question 14:
ncert solutions class 12 maths Miscellaneous Questions Q 14
Solution:
ncert solutions for class 12 maths chapter 7 miscellaneous Q 14
ncert solutions class 12 maths Miscellaneous Questions Q 14 - i

Miscellaneous Exercise Class 11 Maths Question 15:
ncert solutions class 12 maths Miscellaneous Questions Q 15
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 15 - i

Miscellaneous Exercise Class 11 Maths Question 16:
ncert integrals miscellaneous solutions Q 16
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 16 - i

Miscellaneous Exercise Class 11 Maths Question 17:
ncert solutions class 12 maths Miscellaneous Questions Q 17
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 17 - i

Miscellaneous Exercise Class 11 Maths Question 18:
ncert solutions class 12 maths Miscellaneous Questions Q 18
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 18 - i

Miscellaneous Exercise Class 11 Maths Question 20:
ncert solutions class 12 maths Miscellaneous Questions Q 20
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 20 -i
ncert solutions class 12 maths Miscellaneous Questions Q 20 - ii

Miscellaneous Exercise Class 11 Maths Question 21:
ncert solutions class 12 maths Miscellaneous Questions Q 21
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 21 - i

Miscellaneous Exercise Class 11 Maths Question 22:
ncert solutions class 12 maths Miscellaneous Questions Q 22
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 22 - i

Miscellaneous Exercise Class 11 Maths Question 23:
ncert solutions class 12 maths Miscellaneous Questions Q 23
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 23 - i

Miscellaneous Exercise Class 11 Maths Question 24:
ncert solutions class 12 maths Miscellaneous Questions Q 24
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 24 - i
ncert miscellaneous solutions class 12 Q 24 - ii

Miscellaneous Exercise Class 11 Maths Question 25:
ncert solutions class 12 maths Miscellaneous Questions Q 25
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 25 - i

Miscellaneous Exercise Class 11 Maths Question 26:
ncert solutions class 12 maths Miscellaneous Questions Q 26
Solution:
ncert solutions class 12 maths Miscellaneous Questions Q 26 - i

Miscellaneous Exercise Class 11 Maths Question 27:
maths class 12 ncert solutions miscellaneous exercise 27
Solution:
maths class 12 ncert solutions miscellaneous exercise 27 - i
maths class 12 ncert solutions miscellaneous exercise 27 - ii

Miscellaneous Exercise Class 11 Maths Question 28:
maths class 12 ncert solutions miscellaneous exercise 28
Solution:
maths class 12 ncert solutions miscellaneous exercise 28 - i
maths class 12 ncert solutions miscellaneous exercise 28 - ii

Miscellaneous Exercise Class 11 Maths Question 29:
maths class 12 ncert solutions miscellaneous exercise 29
Solution:
maths class 12 ncert solutions miscellaneous exercise 29 - i

Miscellaneous Exercise Class 11 Maths Question 30:
maths class 12 ncert solutions miscellaneous exercise 30
maths class 12 ncert solutions miscellaneous exercise 30 - i

Miscellaneous Exercise Class 11 Maths Question 31:
maths class 12 ncert solutions miscellaneous exercise 31
Solution:
maths class 12 ncert solutions miscellaneous exercise 31 - i

Miscellaneous Exercise Class 11 Maths Question 32:
maths class 12 ncert solutions miscellaneous exercise 32
Solution:
maths class 12 ncert solutions miscellaneous exercise 32- i
maths class 12 ncert solutions miscellaneous exercise 32- ii

Miscellaneous Exercise Class 11 Maths Question 33:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 33
Solution:

Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 33 - ii

Miscellaneous Exercise Class 11 Maths Question 34:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 34
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 34 - i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 34 - ii

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

Miscellaneous Exercise Class 11 Maths Question 36:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 36
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 36 -i

Miscellaneous Exercise Class 11 Maths Question 37:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 37
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 37 - i

Miscellaneous Exercise Class 11 Maths Question 38:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 38
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 38 -i

Miscellaneous Exercise Class 11 Maths Question 39:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 39
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 39-i

Miscellaneous Exercise Class 11 Maths Question 40:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 40
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 40 -i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 40 -ii

Miscellaneous Exercise Class 11 Maths Question 41:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 41
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 41 -i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 41 -ii

Miscellaneous Exercise Class 11 Maths Question 42:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 42
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 42 - i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 42 - ii

Miscellaneous Exercise Class 11 Maths Question 43:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 43
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 43-i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 43-ii

Miscellaneous Exercise Class 11 Maths Question 44:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 44
Solution:
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 44 -i
Integration Class 12 NCERT Solutions Miscellaneous Exercise Q 44 -ii


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