# RD Sharma solutions for Class 12 Maths chapter 19 - Indefinite Integrals [Latest edition]

## Chapter 19: Indefinite Integrals

Exercise 19.1Exercise 19.2Exercise 19.3Exercise 19.4Exercise 19.5Exercise 19.6Exercise 19.7Exercise 19.8Exercise 19.85Exercise 19.9Exercise 19.11Exercise 19..11Exercise 19.12Exercise 19.13Exercise 19.14Exercise 19.15Exercise 19.16Exercise 19.17Exercise 19.18Exercise 19.19Exercise 19.21Exercise 19.22Exercise 19.23Exercise 19.24Exercise 19.25Exercise 19.26Exercise 19.27OthersVery Short AnswersMCQRevision Excercise
Exercise 19.1 [Page 4]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.1 [Page 4]

Exercise 19.1 | Q 1.1 | Page 4

Evaluate of the following integral:

(i)  $\int x^4 dx$

Exercise 19.1 | Q 1.2 | Page 4

Evaluate of the following integral:

$\int x^\frac{5}{4} dx$
Exercise 19.1 | Q 1.3 | Page 4

Evaluate of the following integral:

$\int\frac{1}{x^5}dx$
Exercise 19.1 | Q 1.4 | Page 4

Evaluate of the following integral:

$\int\frac{1}{x^{3/2}}dx$
Exercise 19.1 | Q 1.5 | Page 4

Evaluate of the following integral:

$\int 3^x dx$
Exercise 19.1 | Q 1.6 | Page 4

Evaluate of the following integral:

$\int\frac{1}{\sqrt[3]{x^2}}dx$
Exercise 19.1 | Q 1.7 | Page 4

Evaluate of the following integral:

$\int 3^{2 \log_3} {}^x dx$
Exercise 19.1 | Q 1.8 | Page 4

Evaluate of the following integral:

$\int \log_x \text{x dx}$
Exercise 19.1 | Q 2.1 | Page 4

Evaluate:

$\int\sqrt{\frac{1 + \cos 2x}{2}}dx$
Exercise 19.1 | Q 2.2 | Page 4

Evaluate:

$\int\sqrt{\frac{1 - \cos 2x}{2}}dx$
Exercise 19.1 | Q 3 | Page 4

Evaluate :

$\int\frac{e^{6 \log_e x} - e^{5 \log_e x}}{e^{4 \log_e x} - e^{3 \log_e x}}dx$
Exercise 19.1 | Q 4 | Page 4

Evaluate:

$\int\frac{1}{a^x b^x}dx$
Exercise 19.1 | Q 5.1 | Page 4

Evaluate:

$\int\frac{\cos 2x + 2 \sin^2 x}{\sin^2 x}dx$
Exercise 19.1 | Q 5.2 | Page 4

Evaluate:

$\int\frac{2 \cos^2 x - \cos 2x}{\cos^2 x}dx$
Exercise 19.1 | Q 6 | Page 4

Evaluate:

$\int\frac{e\log \sqrt{x}}{x}dx$
Exercise 19.2 [Pages 14 - 15]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.2 [Pages 14 - 15]

Exercise 19.2 | Q 1 | Page 14
$\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx$
Exercise 19.2 | Q 2 | Page 14
$\int\left( 2^x + \frac{5}{x} - \frac{1}{x^{1/3}} \right)dx$
Exercise 19.2 | Q 3 | Page 14
$\int\left\{ \sqrt{x}\left( a x^2 + bx + c \right) \right\} dx$
Exercise 19.2 | Q 4 | Page 14
$\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx$
Exercise 19.2 | Q 5 | Page 14
$\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx$
Exercise 19.2 | Q 6 | Page 14
$\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx$
Exercise 19.2 | Q 7 | Page 14
$\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx$
Exercise 19.2 | Q 8 | Page 14

$\int\left\{ x^2 + e^{\log x}+ \left( \frac{e}{2} \right)^x \right\} dx$

Exercise 19.2 | Q 9 | Page 14
$\int\left( x^e + e^x + e^e \right) dx$
Exercise 19.2 | Q 10 | Page 14
$\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx$
Exercise 19.2 | Q 11 | Page 14
$\int\frac{1}{\sqrt{x}}\left( 1 + \frac{1}{x} \right) dx$
Exercise 19.2 | Q 12 | Page 14
$\int\frac{x^6 + 1}{x^2 + 1} dx$
Exercise 19.2 | Q 13 | Page 14
$\int\frac{x^{- 1/3} + \sqrt{x} + 2}{\sqrt[3]{x}} dx$
Exercise 19.2 | Q 14 | Page 14
$\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx$
Exercise 19.2 | Q 15 | Page 15

$\int\sqrt{x}\left( 3 - 5x \right) dx$

Exercise 19.2 | Q 16 | Page 15
$\int\frac{\left( x + 1 \right)\left( x - 2 \right)}{\sqrt{x}} dx$
Exercise 19.2 | Q 17 | Page 15
$\int\frac{x^5 + x^{- 2} + 2}{x^2} dx$
Exercise 19.2 | Q 18 | Page 15
$\int \left( 3x + 4 \right)^2 dx$
Exercise 19.2 | Q 19 | Page 15
$\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx$
Exercise 19.2 | Q 20 | Page 15

$\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx$

Exercise 19.2 | Q 21 | Page 15
$\int\frac{\sin^2 x}{1 + \cos x} \text{dx}$
Exercise 19.2 | Q 22 | Page 15
$\int\left( \sec^2 x + {cosec}^2 x \right) dx$
Exercise 19.2 | Q 23 | Page 15
$\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx$
Exercise 19.2 | Q 24 | Page 15
$\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx$
Exercise 19.2 | Q 25 | Page 15
$\int \left( \tan x + \cot x \right)^2 dx$
Exercise 19.2 | Q 26 | Page 15
$\int\frac{1 - \cos 2x}{1 + \cos 2x} dx$
Exercise 19.2 | Q 27 | Page 15

$\int\frac{\cos x}{1 - \cos x} \text{dx }or \int\frac{\cot x}{\text{cosec } {x }- \cot x} dx$
Exercise 19.2 | Q 28 | Page 15
$\int\frac{\cos^2 x - \sin^2 x}{\sqrt{1} + \cos 4x} dx$
Exercise 19.2 | Q 29 | Page 15
$\int\frac{1}{1 - \cos x} dx$
Exercise 19.2 | Q 30 | Page 15
$\int\frac{1}{1 - \sin x} dx$
Exercise 19.2 | Q 31 | Page 15
$\int\frac{\tan x}{\sec x + \tan x} dx$
Exercise 19.2 | Q 32 | Page 15

 ∫  {cosec x} / {"cosec x "- cot x}  dx

Exercise 19.2 | Q 33 | Page 15
$\int\frac{1}{1 + \cos 2x} dx$
Exercise 19.2 | Q 34 | Page 15
$\int\frac{1}{1 - \cos 2x} dx$
Exercise 19.2 | Q 35 | Page 15
$\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx$
Exercise 19.2 | Q 36 | Page 15
$\int \cos^{- 1} \left( \sin x \right) dx$
Exercise 19.2 | Q 37 | Page 15
$\int \cot^{- 1} \left( \frac{\sin 2x}{1 - \cos 2x} \right) dx$
Exercise 19.2 | Q 38 | Page 15
$\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx$
Exercise 19.2 | Q 39 | Page 15
$\int\frac{\left( x^3 + 8 \right)\left( x - 1 \right)}{x^2 - 2x + 4} dx$
Exercise 19.2 | Q 40 | Page 15
$\int \left( a \tan x + b \cot x \right)^2 dx$
Exercise 19.2 | Q 41 | Page 15
$\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx$
Exercise 19.2 | Q 42 | Page 15
$\int\frac{\cos x}{1 + \cos x} dx$
Exercise 19.2 | Q 43 | Page 15
$\int\frac{1 - \cos x}{1 + \cos x} dx$
Exercise 19.2 | Q 45 | Page 15

If f' (x) = x − $\frac{1}{x^2}$  and  f (1)  $\frac{1}{2}, find f(x)$

Exercise 19.2 | Q 46 | Page 15

If f' (x) = x + bf(1) = 5, f(2) = 13, find f(x)

Exercise 19.2 | Q 47 | Page 15

If f' (x) = 8x3 − 2xf(2) = 8, find f(x)

Exercise 19.2 | Q 48 | Page 15

If f' (x) = a sin x + b cos x and f' (0) = 4, f(0) = 3, f

$\left( \frac{\pi}{2} \right)$ = 5, find f(x)

Exercise 19.2 | Q 49 | Page 15
Write the primitive or anti-derivative of
$f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .$

Exercise 19.3 [Pages 23 - 24]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.3 [Pages 23 - 24]

Exercise 19.3 | Q 1 | Page 23
$\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} \text{dx}$
Exercise 19.3 | Q 2 | Page 23
$\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx$
Exercise 19.3 | Q 3 | Page 23
$\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx$
Exercise 19.3 | Q 4 | Page 23
$\int\frac{x + 3}{\left( x + 1 \right)^4} dx$
Exercise 19.3 | Q 5 | Page 23
$\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx$
Exercise 19.3 | Q 6 | Page 23
$\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx$
Exercise 19.3 | Q 7 | Page 23
$\int\frac{2x}{\left( 2x + 1 \right)^2} dx$
Exercise 19.3 | Q 8 | Page 23
$\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx$
Exercise 19.3 | Q 9 | Page 23
$\int\sin x\sqrt{1 + \cos 2x} dx$
Exercise 19.3 | Q 10 | Page 23
$\int\frac{1 + \cos x}{1 - \cos x} dx$
Exercise 19.3 | Q 11 | Page 23
$\int\frac{1 - \cos x}{1 + \cos x} dx$
Exercise 19.3 | Q 12 | Page 23
$\int\frac{1}{1 - \sin\frac{x}{2}} dx$
Exercise 19.3 | Q 13 | Page 23

 ∫  1/ {1+ cos   3x}   dx

Exercise 19.3 | Q 14 | Page 23

$\int \left( e^x + 1 \right)^2 e^x dx$

Exercise 19.3 | Q 15 | Page 23
$\int \left( e^x + 1 \right)^2 e^x dx$
Exercise 19.3 | Q 16 | Page 23
$\int\frac{1 + \cos 4x}{\cot x - \tan x} dx$
Exercise 19.3 | Q 17 | Page 23
$\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx$
Exercise 19.3 | Q 18 | Page 23

$\int \tan^2 \left( 2x - 3 \right) dx$

Exercise 19.3 | Q 19 | Page 24
$\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx$
Exercise 19.4 [Page 30]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.4 [Page 30]

Exercise 19.4 | Q 1 | Page 30

$\int\frac{x^2 + 5x + 2}{x + 2} dx$

Exercise 19.4 | Q 2 | Page 30
$\int\frac{x^3}{x - 2} dx$
Exercise 19.4 | Q 3 | Page 30
$\int\frac{x^2 + x + 5}{3x + 2} dx$
Exercise 19.4 | Q 4 | Page 30
$\int\frac{2x + 3}{\left( x - 1 \right)^2} dx$
Exercise 19.4 | Q 5 | Page 30
$\int\frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} dx$
Exercise 19.4 | Q 6 | Page 30
$\int\frac{2x - 1}{\left( x - 1 \right)^2} dx$
Exercise 19.5 [Page 33]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.5 [Page 33]

Exercise 19.5 | Q 1 | Page 33
$\int\frac{x + 1}{\sqrt{2x + 3}} dx$
Exercise 19.5 | Q 2 | Page 33
∫   x    \sqrt{x + 2}     dx
Exercise 19.5 | Q 3 | Page 33
$\int\frac{x - 1}{\sqrt{x + 4}} dx$
Exercise 19.5 | Q 4 | Page 33
$\int\left( x + 2 \right) \sqrt{3x + 5} \text{dx}$
Exercise 19.5 | Q 5 | Page 33
$\int\frac{2x + 1}{\sqrt{3x + 2}} dx$
Exercise 19.5 | Q 6 | Page 33
$\int\frac{3x + 5}{\sqrt{7x + 9}} dx$
Exercise 19.5 | Q 7 | Page 33
$\int\frac{x}{\sqrt{x + 4}} dx$
Exercise 19.5 | Q 8 | Page 33
$\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx$
Exercise 19.5 | Q 9 | Page 33
$\int\left( 5x + 3 \right) \sqrt{2x - 1} dx$
Exercise 19.5 | Q 10 | Page 33
$\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx$
Exercise 19.6 [Pages 3 - 36]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.6 [Pages 3 - 36]

Exercise 19.6 | Q 1 | Page 36
$\int \text{sin}^2 \left( 2x + 5 \right) \text{dx}$
Exercise 19.6 | Q 2 | Page 36

$\int \sin^3 \left( 2x + 1 \right) \text{dx}$

Exercise 19.6 | Q 3 | Page 36

∫     cos ^4  2x   dx

Exercise 19.6 | Q 4 | Page 36
$\int \sin^2\text{ b x dx}$
Exercise 19.6 | Q 5 | Page 36
$\int \sin^2 \frac{x}{2} dx$
Exercise 19.6 | Q 6 | Page 3
$\int \cos^2 \frac{x}{2} dx$

Exercise 19.6 | Q 7 | Page 36
$\int \cos^2 \text{nx dx}$
Exercise 19.6 | Q 8 | Page 36

 ∫   sin x  \sqrt (1-cos 2x)    dx

Exercise 19.7 [Page 38]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.7 [Page 38]

Exercise 19.7 | Q 1 | Page 38
  ∫  sin 4x cos  7x  dx
Exercise 19.7 | Q 2 | Page 38
 ∫   cos  3x   cos  4x dx
Exercise 19.7 | Q 3 | Page 38
 ∫    cos  mx  cos  nx  dx

Exercise 19.7 | Q 4 | Page 38
$\int\text{sin mx }\text{cos nx dx m }\neq n$
Exercise 19.7 | Q 5 | Page 38

Integrate the following integrals:

$\int\text{sin 2x sin 4x sin 6x dx}$
Exercise 19.7 | Q 6 | Page 38

Integrate the following integrals:

$\int\text { sin x cos 2x sin 3x dx}$
Exercise 19.8, Exercise 19.85 [Pages 47 - 48]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.8, Exercise 19.85 [Pages 47 - 48]

Exercise 19.8 | Q 1 | Page 47
$\int\frac{1}{\sqrt{1 - \cos 2x}} dx$
Exercise 19.8 | Q 2 | Page 47
$\int\frac{1}{\sqrt{1 + \cos x}} dx$
Exercise 19.8 | Q 3 | Page 47
$\int\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} dx$
Exercise 19.8 | Q 4 | Page 47
$\int\sqrt{\frac{1 - \cos x}{1 + \cos x}} dx$
Exercise 19.8 | Q 5 | Page 47

Evaluate the following integrals:

int "sec x"/"sec 2x" "dx"
Exercise 19.8 | Q 6 | Page 47
$\int\frac{\cos 2x}{\left( \cos x + \sin x \right)^2} dx$
Exercise 19.8 | Q 7 | Page 47
$\int\frac{\text{sin} \left( x - a \right)}{\text{sin}\left( x - b \right)} dx$
Exercise 19.8 | Q 8 | Page 47
$\int\frac{\text{sin} \left( x - \alpha \right)}{\text{sin }\left( x + \alpha \right)} dx$
Exercise 19.8 | Q 9 | Page 47
$\int\frac{1 + \tan x}{1 - \tan x} dx$
Exercise 19.8 | Q 10 | Page 47
$\int\frac{\cos x}{\cos \left( x - a \right)} dx$
Exercise 19.8 | Q 11 | Page 47
$\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx$
Exercise 19.8 | Q 12 | Page 47
$\int\frac{e^{3x}}{e^{3x} + 1} dx$
Exercise 19.8 | Q 13 | Page 47
$\int\frac{\sec x \tan x}{3 \sec x + 5} dx$
Exercise 19.8 | Q 14 | Page 47
$\int\frac{1 - \cot x}{1 + \cot x} dx$
Exercise 19.8 | Q 15 | Page 47

 ∫  {sec  x   "cosec " x}/{log  ( tan x) }  dx

Exercise 19.8 | Q 16 | Page 47
$\int\frac{1}{x (3 + \log x)} dx$
Exercise 19.8 | Q 17 | Page 47
$\int\frac{e^x + 1}{e^x + x} dx$
Exercise 19.8 | Q 18 | Page 47
$\int\frac{1}{x \log x} dx$
Exercise 19.8 | Q 19 | Page 47

 ∫  {sin 2x} /{a cos^2  x  + b sin^2  x }   dx

Exercise 19.8 | Q 20 | Page 47
$\int\frac{\cos x}{2 + 3 \sin x} dx$
Exercise 19.8 | Q 21 | Page 47
$\int\frac{1 - \sin x}{x + \cos x} dx$
Exercise 19.8 | Q 22 | Page 47
$\int\frac{a}{b + c e^x} dx$
Exercise 19.8 | Q 23 | Page 47
$\int\frac{1}{e^x + 1} dx$
Exercise 19.8 | Q 24 | Page 47
 ∫ {cot x}/ { log sin x} dx
Exercise 19.8 | Q 25 | Page 47
$\int\frac{e^{2x}}{e^{2x} - 2} dx$
Exercise 19.8 | Q 26 | Page 47
$\int\frac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x} dx$
Exercise 19.8 | Q 27 | Page 48
$\int\frac{\cos 2x + x + 1}{x^2 + \sin 2x + 2x} dx$
Exercise 19.8 | Q 28 | Page 48
$\int\frac{1}{\cos\left( x + a \right) \cos\left( x + b \right)}dx$
Exercise 19.8 | Q 29 | Page 48
$\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx$
Exercise 19.8 | Q 30 | Page 48
$\int\frac{\cos 4x - \cos 2x}{\sin 4x - \sin 2x} dx$
Exercise 19.8 | Q 31 | Page 48
$\int\frac{sec x}{\log \left( \text{sec x }+ \text{tan x} \right)} dx$
Exercise 19.8 | Q 32 | Page 48
 ∫ {"cosec"   x }/ { log  tan   x/2  dx
Exercise 19.8 | Q 33 | Page 48
$\int\frac{1}{ x \text{log x } \text{log }\left( \text{log x }\right)} dx$
Exercise 19.8 | Q 34 | Page 48
$\int\frac{{cosec}^2 x}{1 + \cot x} dx$
Exercise 19.85 | Q 35 | Page 48
$\int\frac{10 x^9 + {10}^x \log_e 10}{{10}^x + x^{10}} dx$
Exercise 19.8 | Q 36 | Page 48
$\int\frac{1 - \sin 2x}{x + \cos^2 x} dx$
Exercise 19.8 | Q 37 | Page 48
 ∫  {1+tan}/{ x + log  sec  x   dx}
Exercise 19.8 | Q 38 | Page 48
$\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x} dx$
Exercise 19.8 | Q 39 | Page 48
$\int\frac{x + 1}{x \left( x + \log x \right)} dx$
Exercise 19.8 | Q 40 | Page 48
$\int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)} dx$
Exercise 19.8 | Q 41 | Page 48
$\int\frac{\sec^2 x}{\tan x + 2} dx$
Exercise 19.8 | Q 42 | Page 48
$\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx$
Exercise 19.8 | Q 43 | Page 48
$\int\frac{\sin 2x}{\sin 5x \sin 3x} dx$
Exercise 19.8 | Q 44 | Page 48
$\int\frac{1 + \cot x}{x + \log \sin x} dx$
Exercise 19.8 | Q 45 | Page 48
$\int\frac{1}{\sqrt{x}\left( \sqrt{x} + 1 \right)} dx$
Exercise 19.8 | Q 46 | Page 48
 ∫  tan 2x tan 3x  tan 5x    dx
Exercise 19.8 | Q 47 | Page 48
$\int\left\{ 1 + \tan x \tan \left( x + \theta \right) \right\} dx$
Exercise 19.8 | Q 48 | Page 48
$\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx$
Exercise 19.8 | Q 49 | Page 48
$\int\frac{e^{x - 1} + x^{e - 1}}{e^x + x^e} dx$
Exercise 19.8 | Q 50 | Page 48
$\int\frac{1}{\sin x \cos^2 x} dx$
Exercise 19.8 | Q 51 | Page 48
$\int\frac{1}{\cos 3x - \cos x} dx$
Exercise 19.9 [Pages 57 - 59]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.9 [Pages 57 - 59]

Exercise 19.9 | Q 1 | Page 57
 ∫  log x / x  dx

Exercise 19.9 | Q 2 | Page 57
$\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx$
Exercise 19.9 | Q 3 | Page 57
$\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx$
Exercise 19.9 | Q 4 | Page 57
$\int\sqrt{1 + e^x} . e^x dx$
Exercise 19.9 | Q 5 | Page 57

  =  ∫ root (3){ cos^2 x}  sin x   dx

Exercise 19.9 | Q 6 | Page 57
$\int\frac{e^x}{\left( 1 + e^x \right)^2} dx$
Exercise 19.9 | Q 7 | Page 57

 ∫       cot^3   x  "cosec"^2   x   dx

Exercise 19.9 | Q 8 | Page 57

$\int\frac{\left\{ e^{\sin^{- 1} }x \right\}^2}{\sqrt{1 - x^2}} dx$

Exercise 19.9 | Q 9 | Page 57
$\int\frac{1 + \sin x}{\sqrt{x - \cos x}} dx$
Exercise 19.9 | Q 10 | Page 57

$\int\frac{1}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2} dx$

Exercise 19.9 | Q 11 | Page 58

$\int\frac{\cot x}{\sqrt{\sin x}} dx$

Exercise 19.9 | Q 12 | Page 58
$\int\frac{\tan x}{\sqrt{\cos x}} dx$
Exercise 19.9 | Q 13 | Page 58
$\int\frac{\cos^3 x}{\sqrt{\sin x}} dx$
Exercise 19.9 | Q 14 | Page 58
$\int\frac{\sin^3 x}{\sqrt{\cos x}} dx$
Exercise 19.9 | Q 15 | Page 58
$\int\frac{1}{\sqrt{\tan^{- 1} x} . \left( 1 + x^2 \right)} dx$
Exercise 19.9 | Q 16 | Page 58

$\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx$

Exercise 19.9 | Q 17 | Page 58

$\int\frac{1}{x} \left( \log x \right)^2 dx$

Exercise 19.9 | Q 18 | Page 58
$\int \sin^5\text{ x }\text{cos x dx}$
Exercise 19.9 | Q 19 | Page 58
$\int \tan^{3/2} x \sec^2 \text{x dx}$
Exercise 19.9 | Q 20 | Page 58
$\int\frac{x^3}{\left( x^2 + 1 \right)^3} dx$
Exercise 19.9 | Q 21 | Page 58
$\int\left( 4x + 2 \right)\sqrt{x^2 + x + 1} \text{dx}$
Exercise 19.9 | Q 22 | Page 58
$\int\frac{4x + 3}{\sqrt{2 x^2 + 3x + 1}} dx$
Exercise 19.9 | Q 23 | Page 58
$\int\frac{1}{1 + \sqrt{x}} dx$
Exercise 19.9 | Q 24 | Page 58
$\int e^{cos^2 x} \text{sin 2x dx}$
Exercise 19.9 | Q 25 | Page 58
$\int\frac{1 + \cos x}{\left( x + \sin x \right)^3} dx$
Exercise 19.9 | Q 26 | Page 58
$\int\frac{\cos x - \sin x}{1 + \sin 2x} dx$
Exercise 19.9 | Q 27 | Page 58
$\int\frac{\sin 2x}{\left( a + b \cos 2x \right)^2} dx$
Exercise 19.9 | Q 28 | Page 58
$\int\frac{\log x^2}{x} dx$
Exercise 19.9 | Q 29 | Page 58
$\int\frac{\sin x}{\left( 1 + \cos x \right)^2} dx$

Exercise 19.9 | Q 30 | Page 58
$\int\cot x \cdot \log \text{sin x dx}$
Exercise 19.9 | Q 31 | Page 58
$\int\sec x \cdot \text{log} \left( \sec x + \tan x \right) dx$
Exercise 19.9 | Q 32 | Page 58
$\text{ ∫ cosec x log} \left( \text{cosec x} - \cot x \right) dx$
Exercise 19.9 | Q 33 | Page 58
$\int x^3 \cos x^4 dx$
Exercise 19.9 | Q 34 | Page 58
$\int x^3 \sin x^4 dx$
Exercise 19.9 | Q 35 | Page 58
$\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx$
Exercise 19.9 | Q 36 | Page 58
$\int x^3 \sin \left( x^4 + 1 \right) dx$
Exercise 19.9 | Q 37 | Page 58
$\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x e^x \right)} dx$
Exercise 19.9 | Q 38 | Page 58
$\int x^2 e^{x^3} \cos \left( e^{x^3} \right) dx$
Exercise 19.9 | Q 39 | Page 58
$\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx$
Exercise 19.9 | Q 40 | Page 58
$\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx$
Exercise 19.9 | Q 41 | Page 58
   ∫     tan x    .  sec^2 x   \sqrt{1 - tan^2 x}     dx\
Exercise 19.9 | Q 42 | Page 58
$\int\log x\frac{\text{sin} \left\{ 1 + \left( \log x \right)^2 \right\}}{x} dx$
Exercise 19.9 | Q 43 | Page 58
$\int\frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) dx$
Exercise 19.9 | Q 44 | Page 58
$\int \sec^4 \text{ x tan x dx}$
Exercise 19.9 | Q 45 | Page 58
$\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx$
Exercise 19.9 | Q 46 | Page 58
$\int\frac{\cos^5 x}{\sin x} dx$
Exercise 19.9 | Q 47 | Page 59
$\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx$
Exercise 19.9 | Q 48 | Page 59
$\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( \text{x e}^x \right)} dx$
Exercise 19.9 | Q 49 | Page 59
$\int 5^{x + \tan^{- 1} x} . \left( \frac{x^2 + 2}{x^2 + 1} \right) dx$
Exercise 19.9 | Q 50 | Page 59

 ∫   e^{m   sin ^-1  x}/ \sqrt{1-x^2}   dx

Exercise 19.9 | Q 51 | Page 59
$\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx$
Exercise 19.9 | Q 52 | Page 59
$\int\frac{\sin \left( \tan^{- 1} x \right)}{1 + x^2} dx$
Exercise 19.9 | Q 53 | Page 59
$\int\frac{\sin \left( \text{log x} \right)}{x} dx$
Exercise 19.9 | Q 54 | Page 59
$\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx$
Exercise 19.9 | Q 55 | Page 59
$\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx$
Exercise 19.9 | Q 56 | Page 59
 ∫    x   {tan^{- 1} x^2}/{1 + x^4} dx
Exercise 19.9 | Q 57 | Page 59
$\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx$

Exercise 19.9 | Q 58 | Page 59
$\int\frac{\text{sin }\left( \text{2 + 3 log x }\right)}{x} dx$
Exercise 19.9 | Q 59 | Page 59
$\ ∫ x \text{ e}^{x^2} dx$
Exercise 19.9 | Q 60 | Page 59
$\int\frac{e^{2x}}{1 + e^x} dx$
Exercise 19.9 | Q 61 | Page 59
$\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx$
Exercise 19.9 | Q 62 | Page 59
$\int \tan^3 \text{2x sec 2x dx}$
Exercise 19.9 | Q 63 | Page 59
$\int\frac{x + \sqrt{x + 1}}{x + 2} dx$
Exercise 19.9 | Q 64 | Page 59
$\int 5^{5^{5^x}} 5^{5^x} 5^x dx$
Exercise 19.9 | Q 65 | Page 59
$\int\frac{1}{x\sqrt{x^4 - 1}} dx$
Exercise 19.9 | Q 66 | Page 59
$\int\sqrt {e^x- 1} \text{dx}$
Exercise 19.9 | Q 67 | Page 59
$\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx$
Exercise 19.9 | Q 68 | Page 59
$\int\frac{x^5}{\sqrt{1 + x^3}} dx$
Exercise 19.9 | Q 69 | Page 59
$\int4 x^3 \sqrt{5 - x^2} dx$
Exercise 19.9 | Q 70 | Page 59
$\int\frac{1}{\sqrt{x} + x} \text{ dx }$
Exercise 19.9 | Q 71 | Page 59
$\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx$
Exercise 19.9 | Q 72 | Page 59
$\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }$
Exercise 19.1 [Page 65]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.1 [Page 65]

Exercise 19.1 | Q 1 | Page 65
$\int x^2 \sqrt{x + 2} \text{ dx }$
Exercise 19.1 | Q 2 | Page 65
$\int\frac{x^2}{\sqrt{x - 1}} dx$
Exercise 19.1 | Q 3 | Page 65
$\int\frac{x^2}{\sqrt{3x + 4}} dx$
Exercise 19.1 | Q 4 | Page 65
$\int\frac{2x - 1}{\left( x - 1 \right)^2} dx$
Exercise 19.1 | Q 5 | Page 65
$\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }$
Exercise 19.1 | Q 6 | Page 65
$\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx$
Exercise 19.1 | Q 7 | Page 65
$\int\frac{x^2}{\sqrt{1 - x}} dx$
Exercise 19.1 | Q 8 | Page 65
$\ \int\ x \left( 1 - x \right)^{23} dx$

Exercise 19.1 | Q 9 | Page 65
$\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx$
Exercise 19.1 | Q 10 | Page 65

 ∫   1 /{x^{1/3} ( x^{1/3} -1)}    dx

Exercise 19.11, Exercise 19..11 [Page 69]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.11, Exercise 19..11 [Page 69]

Exercise 19.11 | Q 1 | Page 69
 ∫  tan^3    x   sec^2  x   dx
Exercise 19.11 | Q 2 | Page 69

 ∫   tan   x   sec^4  x   dx

Exercise 19.11 | Q 3 | Page 69
 ∫  tan^5 x   sec ^4 x   dx
Exercise 19.11 | Q 4 | Page 69
 ∫  sec^6   x  tan    x   dx
Exercise 19.11 | Q 5 | Page 69

 ∫      tan^5    x   dx

Exercise 19.11 | Q 6 | Page 69

 ∫    \sqrt{tan x}     sec^4  x   dx

Exercise 19.11 | Q 7 | Page 69
$\int \sec^4 2x \text{ dx }$
Exercise 19.11 | Q 8 | Page 69
$\int {cosec}^4 \text{ 3x } \text{ dx }$
Exercise 19..11 | Q 9 | Page 69
$\int \cot^n {cosec}^2 \text{ x dx } , n \neq - 1$
Exercise 19.11 | Q 10 | Page 69
$\int \cot^5 \text{ x } {cosec}^4 x\text{ dx }$
Exercise 19.11 | Q 11 | Page 69
$\int \cot^5 x \text{ dx }$
Exercise 19.11 | Q 12 | Page 69
$\int \cot^6 x \text{ dx }$
Exercise 19.12 [Page 73]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.12 [Page 73]

Exercise 19.12 | Q 1 | Page 73
$\int \sin^4 x \cos^3 x \text{ dx }$
Exercise 19.12 | Q 2 | Page 73
$\int \sin^5 x \text{ dx }$
Exercise 19.12 | Q 3 | Page 73
$\int \cos^5 x \text{ dx }$
Exercise 19.12 | Q 4 | Page 73
$\int \sin^5 x \cos x \text{ dx }$
Exercise 19.12 | Q 5 | Page 73
$\int \sin^3 x \cos^6 x \text{ dx }$
Exercise 19.12 | Q 6 | Page 73
$\int \cos^7 x \text{ dx }$
Exercise 19.12 | Q 7 | Page 73
$\int x \cos^3 x^2 \sin x^2 \text{ dx }$
Exercise 19.12 | Q 8 | Page 73
$\int \sin^7 x \text{ dx }$
Exercise 19.12 | Q 9 | Page 73
$\int \sin^3 x \cos^5 x \text{ dx }$
Exercise 19.12 | Q 10 | Page 73
$\int\frac{1}{\sin^4 x \cos^2 x} dx$
Exercise 19.12 | Q 11 | Page 73
$\int\frac{1}{\sin^3 x \cos^5 x} dx$
Exercise 19.12 | Q 12 | Page 73

 = ∫1/{sin^3 x cos^ 2x} dx

Exercise 19.12 | Q 13 | Page 73
$\int\frac{1}{\sin x \cos^3 x} dx$
Exercise 19.13 [Page 79]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.13 [Page 79]

Exercise 19.13 | Q 1 | Page 79
Evaluate the following integrals:
$\int\frac{x^2}{\left( a^2 - x^2 \right)^{3/2}}dx$
Exercise 19.13 | Q 2 | Page 79

Evaluate the following integrals:

$\int\frac{x^7}{\left( a^2 - x^2 \right)^5}dx$
Exercise 19.13 | Q 3 | Page 79

Evaluate the following integrals:

$\int\cos\left\{ 2 \cot^{- 1} \sqrt{\frac{1 + x}{1 - x}} \right\}dx$
Exercise 19.13 | Q 4 | Page 79

Evaluate the following integrals:

$\int\frac{\sqrt{1 + x^2}}{x^4}dx$
Exercise 19.13 | Q 5 | Page 79

Evaluate the following integrals:

$\int\frac{1}{\left( x^2 + 2x + 10 \right)^2}dx$

Exercise 19.14 [Page 83]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.14 [Page 83]

Exercise 19.14 | Q 1 | Page 83
$\int\frac{1}{a^2 - b^2 x^2} dx$
Exercise 19.14 | Q 2 | Page 83
 ∫  {1}/{a^2 x^2- b^2}dx
Exercise 19.14 | Q 3 | Page 83
$\int\frac{1}{a^2 x^2 + b^2} dx$
Exercise 19.14 | Q 4 | Page 83
$\int\frac{x^2 - 1}{x^2 + 4} dx$
Exercise 19.14 | Q 5 | Page 83
$\int\frac{1}{\sqrt{1 + 4 x^2}} dx$

Exercise 19.14 | Q 6 | Page 83
$\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx$
Exercise 19.14 | Q 7 | Page 83
$\int\frac{1}{\sqrt{a^2 - b^2 x^2}} dx$
Exercise 19.14 | Q 8 | Page 83
$\int\frac{1}{\sqrt{\left( 2 - x \right)^2 + 1}} dx$
Exercise 19.14 | Q 9 | Page 83
$\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx$
Exercise 19.14 | Q 10 | Page 83
$\int\frac{x^4 + 1}{x^2 + 1} dx$
Exercise 19.15 [Page 86]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.15 [Page 86]

Exercise 19.15 | Q 1 | Page 86
$\int\frac{1}{4 x^2 + 12x + 5} dx$
Exercise 19.15 | Q 2 | Page 86
$\int\frac{1}{x^2 - 10x + 34} dx$
Exercise 19.15 | Q 3 | Page 86
$\int\frac{1}{1 + x - x^2} \text{ dx }$
Exercise 19.15 | Q 4 | Page 86
$\int\frac{1}{2 x^2 - x - 1} dx$
Exercise 19.15 | Q 5 | Page 86
$\int\frac{1}{x^2 + 6x + 13} dx$
Exercise 19.16 [Page 90]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.16 [Page 90]

Exercise 19.16 | Q 1 | Page 90
$\int\frac{\sec^2 x}{1 - \tan^2 x} dx$
Exercise 19.16 | Q 2 | Page 90
$\int\frac{e^x}{1 + e^{2x}} dx$
Exercise 19.16 | Q 3 | Page 90
$\int\frac{\cos x}{\sin^2 x + 4 \sin x + 5} dx$
Exercise 19.16 | Q 4 | Page 90
$\int\frac{e^x}{e^{2x} + 5 e^x + 6} dx$
Exercise 19.16 | Q 5 | Page 90
$\int\frac{e^{3x}}{4 e^{6x} - 9} dx$
Exercise 19.16 | Q 6 | Page 90
$\int\frac{dx}{e^x + e^{- x}}$
Exercise 19.16 | Q 7 | Page 90
$\int\frac{x}{x^4 + 2 x^2 + 3} dx$
Exercise 19.16 | Q 8 | Page 90
$\int\frac{3 x^5}{1 + x^{12}} dx$
Exercise 19.16 | Q 9 | Page 90
 ∫  { x^2 dx}/{x^6 - a^6} dx
Exercise 19.16 | Q 10 | Page 90
$\int\frac{x^2}{x^6 + a^6} dx$
Exercise 19.16 | Q 11 | Page 90
$\int\frac{1}{x \left( x^6 + 1 \right)} dx$
Exercise 19.16 | Q 12 | Page 90
$\int\frac{x}{x^4 - x^2 + 1} dx$
Exercise 19.16 | Q 13 | Page 90
$\int\frac{x}{3 x^4 - 18 x^2 + 11} dx$
Exercise 19.16 | Q 14 | Page 90
$\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx$
Exercise 19.16 | Q 15 | Page 90
  ∫    {1} / {cos x  + "cosec x" } dx
Exercise 19.17 [Page 93]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.17 [Page 93]

Exercise 19.17 | Q 1 | Page 93
$\int\frac{1}{\sqrt{2x - x^2}} dx$
Exercise 19.17 | Q 2 | Page 93
$\int\frac{1}{\sqrt{8 + 3x - x^2}} dx$
Exercise 19.17 | Q 3 | Page 93
$\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx$
Exercise 19.17 | Q 4 | Page 93
$\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx$
Exercise 19.17 | Q 5 | Page 93
$\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)$
Exercise 19.17 | Q 6 | Page 93
$\int\frac{1}{\sqrt{7 - 3x - 2 x^2}} dx$
Exercise 19.17 | Q 7 | Page 93
$\int\frac{1}{\sqrt{16 - 6x - x^2}} dx$
Exercise 19.17 | Q 8 | Page 93
$\int\frac{1}{\sqrt{7 - 6x - x^2}} dx$
Exercise 19.17 | Q 9 | Page 93
$\int\frac{1}{\sqrt{5 x^2 - 2x}} dx$
Exercise 19.18 [Pages 98 - 99]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.18 [Pages 98 - 99]

Exercise 19.18 | Q 1 | Page 98
$\int\frac{x}{\sqrt{x^4 + a^4}} dx$
Exercise 19.18 | Q 2 | Page 98
$\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx$
Exercise 19.18 | Q 3 | Page 98
$\int\frac{e^x}{\sqrt{16 - e^{2x}}} dx$
Exercise 19.18 | Q 4 | Page 99
$\int\frac{\cos x}{\sqrt{4 + \sin^2 x}} dx$
Exercise 19.18 | Q 5 | Page 99
$\int\frac{\sin x}{\sqrt{4 \cos^2 x - 1}} dx$
Exercise 19.18 | Q 6 | Page 99
$\int\frac{x}{\sqrt{4 - x^4}} dx$
Exercise 19.18 | Q 7 | Page 99
$\int\frac{1}{x\sqrt{4 - 9 \left( \log x \right)^2}} dx$
Exercise 19.18 | Q 8 | Page 99
$\int\frac{\sin 8x}{\sqrt{9 + \sin^4 4x}} dx$
Exercise 19.18 | Q 9 | Page 99
$\int\frac{\cos 2x}{\sqrt{\sin^2 2x + 8}} dx$
Exercise 19.18 | Q 10 | Page 99
$\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx$
Exercise 19.18 | Q 11 | Page 99
$\int\frac{\sin 2x}{\sqrt{\cos^4 x - \sin^2 x + 2}} dx$
Exercise 19.18 | Q 12 | Page 99
$\int\frac{\cos x}{\sqrt{4 - \sin^2 x}} dx$
Exercise 19.18 | Q 13 | Page 99
$\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx$
Exercise 19.18 | Q 14 | Page 99
$\int\frac{1}{\sqrt{\left( 1 - x^2 \right)\left\{ 9 + \left( \sin^{- 1} x \right)^2 \right\}}} dx$
Exercise 19.18 | Q 15 | Page 99
$\int\frac{\cos x}{\sqrt{\sin^2 x - 2 \sin x - 3}} dx$
Exercise 19.18 | Q 16 | Page 99
  ∫ \sqrt{"cosec x"- 1}  dx
Exercise 19.18 | Q 17 | Page 99
$\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} dx$
Exercise 19.18 | Q 18 | Page 99
$\int\frac{\cos x - \sin x}{\sqrt{8 - \sin2x}}dx$
Exercise 19.19 [Page 104]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.19 [Page 104]

Exercise 19.19 | Q 1 | Page 104
$\int\frac{x}{x^2 + 3x + 2} dx$
Exercise 19.19 | Q 2 | Page 104
$\int\frac{x + 1}{x^2 + x + 3} dx$
Exercise 19.19 | Q 3 | Page 104

 ∫  {x-3} /{ x^2 + 2x - 4 } dx

Exercise 19.19 | Q 4 | Page 104
$\int\frac{2x - 3}{x^2 + 6x + 13} dx$
Exercise 19.19 | Q 5 | Page 104
$\int\frac{x - 1}{3 x^2 - 4x + 3} dx$
Exercise 19.19 | Q 6 | Page 104
$\int\frac{2x}{2 + x - x^2} \text{ dx }$
Exercise 19.19 | Q 7 | Page 104
$\int\frac{1 - 3x}{3 x^2 + 4x + 2}\text{ dx}$
Exercise 19.19 | Q 8 | Page 104
$\int\frac{2x + 5}{x^2 - x - 2} \text{ dx }$
Exercise 19.19 | Q 9 | Page 104
$\int\frac{a x^3 + bx}{x^4 + c^2} dx$
Exercise 19.19 | Q 10 | Page 104
$\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx$
Exercise 19.19 | Q 11 | Page 104
$\int\frac{x + 2}{2 x^2 + 6x + 5}\text{ dx }$
Exercise 19.19 | Q 12 | Page 104

Evaluate the following integrals:

$\int\frac{5x - 2}{1 + 2x + 3 x^2} \text{ dx }$
Exercise 19.19 | Q 13 | Page 104
$\int\frac{x + 5}{3 x^2 + 13x - 10}\text{ dx }$
Exercise 19.19 | Q 14 | Page 104
$\int\frac{\left( 3\sin x - 2 \right)\cos x}{13 - \cos^2 x - 7\sin x}dx$
Exercise 19.19 | Q 15 | Page 104
$\int\frac{x + 7}{3 x^2 + 25x + 28}\text{ dx}$
Exercise 19.19 | Q 16 | Page 104
$\int\frac{x^3}{x^4 + x^2 + 1}dx$
Exercise 19.19 | Q 17 | Page 104
$\int\frac{x^3 - 3x}{x^4 + 2 x^2 - 4}dx$
Exercise 19.2 [Page 106]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.2 [Page 106]

Exercise 19.2 | Q 1 | Page 106
$\int\frac{x^2 + x + 1}{x^2 - x} dx$
Exercise 19.2 | Q 2 | Page 106
$\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }$
Exercise 19.2 | Q 3 | Page 106
$\int\frac{\left( 1 - x^2 \right)}{x \left( 1 - 2x \right)} \text {dx$
Exercise 19.2 | Q 4 | Page 106
$\int\frac{x^2 + 1}{x^2 - 5x + 6} \text{ dx }$

Exercise 19.2 | Q 5 | Page 106
$\int\frac{x^2}{x^2 + 7x + 10} dx$
Exercise 19.2 | Q 5 | Page 106
$\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }$
Exercise 19.2 | Q 6 | Page 106
$\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }$
Exercise 19.2 | Q 7 | Page 106
$\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx$
Exercise 19.2 | Q 8 | Page 106
$\int\frac{x^3 + x^2 + 2x + 1}{x^2 - x + 1}\text{ dx }$
Exercise 19.2 | Q 9 | Page 106
$\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }$
Exercise 19.2 | Q 10 | Page 106
$\int\frac{x^2}{x^2 + 6x + 12} \text{ dx }$
Exercise 19.21 [Pages 110 - 111]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.21 [Pages 110 - 111]

Exercise 19.21 | Q 1 | Page 110
$\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }$
Exercise 19.21 | Q 2 | Page 110
$\int\frac{2x + 1}{\sqrt{x^2 + 2x - 1}}\text{ dx }$
Exercise 19.21 | Q 3 | Page 110
$\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} \text{ dx }$
Exercise 19.21 | Q 4 | Page 110
$\int\frac{6x - 5}{\sqrt{3 x^2 - 5x + 1}} \text{ dx }$
Exercise 19.21 | Q 5 | Page 110

$\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }$

Exercise 19.21 | Q 6 | Page 110

$\int\frac{x}{\sqrt{8 + x - x^2}} dx$

Exercise 19.21 | Q 7 | Page 110
$\int\frac{x + 2}{\sqrt{x^2 + 2x - 1}} \text{ dx }$
Exercise 19.21 | Q 8 | Page 110
$\int\frac{x + 2}{\sqrt{x^2 - 1}} \text{ dx }$
Exercise 19.21 | Q 9 | Page 110
$\int\frac{x - 1}{\sqrt{x^2 + 1}} \text{ dx }$
Exercise 19.21 | Q 10 | Page 110
$\int\frac{x}{\sqrt{x^2 + x + 1}} \text{ dx }$
Exercise 19.21 | Q 11 | Page 110
$\int\frac{x + 1}{\sqrt{x^2 + 1}} dx$
Exercise 19.21 | Q 12 | Page 110
$\int\frac{2x + 5}{\sqrt{x^2 + 2x + 5}} dx$
Exercise 19.21 | Q 13 | Page 110
$\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }$
Exercise 19.21 | Q 14 | Page 110
$\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }$
Exercise 19.21 | Q 15 | Page 111
$\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }$
Exercise 19.21 | Q 16 | Page 111
$\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }$
Exercise 19.21 | Q 17 | Page 111
$\int\frac{5x + 3}{\sqrt{x^2 + 4x + 10}} \text{ dx }$
Exercise 19.21 | Q 18 | Page 111

Evaluate the following integrals:

$\int\frac{x + 2}{\sqrt{x^2 + 2x + 3}} \text{ dx }$
Exercise 19.22 [Page 114]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.22 [Page 114]

Exercise 19.22 | Q 1 | Page 114
$\int\frac{1}{4 \cos^2 x + 9 \sin^2 x}\text{ dx }$
Exercise 19.22 | Q 2 | Page 114
$\int\frac{1}{4 \sin^2 x + 5 \cos^2 x} \text{ dx }$
Exercise 19.22 | Q 3 | Page 114
$\int\frac{2}{2 + \sin 2x}\text{ dx }$
Exercise 19.22 | Q 4 | Page 114
$\int\frac{\cos x}{\cos 3x} \text{ dx }$
Exercise 19.22 | Q 5 | Page 114
$\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }$
Exercise 19.22 | Q 6 | Page 114
$\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }$
Exercise 19.22 | Q 7 | Page 114
$\int\frac{1}{\left( \sin x - 2 \cos x \right)\left( 2 \sin x + \cos x \right)} \text{ dx }$
Exercise 19.22 | Q 8 | Page 114
$\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }$
Exercise 19.22 | Q 9 | Page 114
$\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx$
Exercise 19.22 | Q 10 | Page 114
$\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }$
Exercise 19.22 | Q 11 | Page 114
$\int\frac{1}{\cos 2x + 3 \sin^2 x} dx$
Exercise 19.23 [Page 117]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.23 [Page 117]

Exercise 19.23 | Q 1 | Page 117
$\int\frac{1}{5 + 4 \cos x} dx$
Exercise 19.23 | Q 2 | Page 117
$\int\frac{1}{5 - 4 \sin x} \text{ dx }$
Exercise 19.23 | Q 3 | Page 117
$\int\frac{1}{1 - 2 \sin x} \text{ dx }$
Exercise 19.23 | Q 4 | Page 117
$\int\frac{1}{4 \cos x - 1} \text{ dx }$
Exercise 19.23 | Q 5 | Page 117
$\int\frac{1}{1 - \sin x + \cos x} \text{ dx }$
Exercise 19.23 | Q 6 | Page 117
$\int\frac{1}{3 + 2 \sin x + \cos x} \text{ dx }$
Exercise 19.23 | Q 7 | Page 117
$\int\frac{1}{13 + 3 \cos x + 4 \sin x} dx$
Exercise 19.23 | Q 8 | Page 117
$\int\frac{1}{\cos x - \sin x} dx$
Exercise 19.23 | Q 9 | Page 117
$\int\frac{1}{\sin x + \cos x} \text{ dx }$
Exercise 19.23 | Q 10 | Page 117
$\int\frac{1}{5 - 4 \cos x} \text{ dx }$
Exercise 19.23 | Q 11 | Page 117
$\int\frac{1}{2 + \sin x + \cos x} \text{ dx }$
Exercise 19.23 | Q 12 | Page 117
$\int\frac{1}{\sin x + \sqrt{3} \cos x} \text{ dx }$
Exercise 19.23 | Q 13 | Page 117
$\int\frac{1}{\sqrt{3} \sin x + \cos x} dx$
Exercise 19.23 | Q 14 | Page 117
$\int\frac{1}{\sin x - \sqrt{3} \cos x} \text{ dx }$
Exercise 19.23 | Q 15 | Page 117
$\int\frac{1}{5 + 7 \cos x + \sin x} dx$
Exercise 19.24 [Page 122]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.24 [Page 122]

Exercise 19.24 | Q 1 | Page 122
$\int\frac{1}{1 - \cot x} dx$
Exercise 19.24 | Q 2 | Page 122
$\int\frac{1}{1 - \tan x} \text{ dx }$
Exercise 19.24 | Q 3 | Page 122
$\int\frac{3 + 2 \cos x + 4 \sin x}{2 \sin x + \cos x + 3} \text{ dx }$
Exercise 19.24 | Q 4 | Page 122
$\int\frac{1}{p + q \tan x} \text{ dx }$
Exercise 19.24 | Q 5 | Page 122
$\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }$
Exercise 19.24 | Q 6 | Page 122
$\int\frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx$
Exercise 19.24 | Q 7 | Page 122
$\int\frac{1}{3 + 4 \cot x} dx$
Exercise 19.24 | Q 8 | Page 122
$\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }$
Exercise 19.24 | Q 9 | Page 122
$\int\frac{1}{4 + 3 \tan x} dx$
Exercise 19.24 | Q 10 | Page 122
$\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }$
Exercise 19.24 | Q 11 | Page 122
$\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }$
Exercise 19.25 [Pages 133 - 134]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.25 [Pages 133 - 134]

Exercise 19.25 | Q 1 | Page 133
$\int x \cos x\ dx$
Exercise 19.25 | Q 2 | Page 133
$\int\text{ log }\left( x + 1 \right) \text{ dx }$
Exercise 19.25 | Q 3 | Page 133
$\int x^3 \text{ log x dx }$
Exercise 19.25 | Q 4 | Page 133
$\int x e^x \text{ dx }$
Exercise 19.25 | Q 5 | Page 133
$\int x e^{2x} \text{ dx }$
Exercise 19.25 | Q 6 | Page 133
$\int x^2 e^{- x} \text{ dx }$
Exercise 19.25 | Q 7 | Page 133
$\int x^2 \text{ cos x dx }$
Exercise 19.25 | Q 8 | Page 133
$\int x^2 \cos 2x\ \text{ dx }$
Exercise 19.25 | Q 9 | Page 133
$\int x \text{ sin 2x dx }$
Exercise 19.25 | Q 10 | Page 133
$\int\frac{\log \left( \log x \right)}{x} dx$
Exercise 19.25 | Q 11 | Page 133
$\int x^2 \text{ cos x dx }$
Exercise 19.25 | Q 12 | Page 133

$\int x\ {cosec}^2 \text{ x }\ \text{ dx }$

Exercise 19.25 | Q 13 | Page 133
$\int x \cos^2 x\ dx$
Exercise 19.25 | Q 14 | Page 133
int"x"^"n"."log"  "x"  "dx"
Exercise 19.25 | Q 15 | Page 133
$\int\frac{\log x}{x^n}\text{ dx }$
Exercise 19.25 | Q 16 | Page 133
$\int x^2 \sin^2 x\ dx$
Exercise 19.25 | Q 17 | Page 133
$\int2 x^3 e^{x^2} dx$
Exercise 19.25 | Q 18 | Page 133
$\int x^3 \cos x^2 dx$
Exercise 19.25 | Q 19 | Page 133
$\int x \sin x \cos x\ dx$

Exercise 19.25 | Q 20 | Page 133
 ∫    sin x log  (\text{ cos x ) } dx
Exercise 19.25 | Q 21 | Page 133
$\int \left( \log x \right)^2 \cdot x\ dx$
Exercise 19.25 | Q 22 | Page 133
$\int e^\sqrt{x} \text{ dx }$
Exercise 19.25 | Q 23 | Page 133
$\int\frac{\text{ log }\left( x + 2 \right)}{\left( x + 2 \right)^2} \text{ dx }$
Exercise 19.25 | Q 24 | Page 133
$\int\frac{x + \sin x}{1 + \cos x} \text{ dx }$
Exercise 19.25 | Q 25 | Page 133
$\int \log_{10} x\ dx$
Exercise 19.25 | Q 26 | Page 133
$\int\cos\sqrt{x}\ dx$
Exercise 19.25 | Q 27 | Page 133

Evaluate the following integrals:

$\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx$

Exercise 19.25 | Q 28 | Page 133

Evaluate the following integrals:

$\int\frac{\log x}{\left( x + 1 \right)^2}dx$

Exercise 19.25 | Q 29 | Page 133
$\int {cosec}^3 x\ dx$
Exercise 19.25 | Q 30 | Page 133
$\int \sec^{- 1} \sqrt{x}\ dx$
Exercise 19.25 | Q 31 | Page 134
$\int \sin^{- 1} \sqrt{x} \text{ dx }$
Exercise 19.25 | Q 32 | Page 134

 ∫  x tan ^2 x dx
Exercise 19.25 | Q 33 | Page 134
$\int x\left( \frac{\sec 2x - 1}{\sec 2x + 1} \right) dx$
Exercise 19.25 | Q 34 | Page 134
$\int\left( x + 1 \right) \text{ e}^x \text{ log } \left( x e^x \right) dx$
Exercise 19.25 | Q 35 | Page 134
$\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx }$
Exercise 19.25 | Q 36 | Page 134
$\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ dx }$
Exercise 19.25 | Q 37 | Page 134
$\int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx$
Exercise 19.25 | Q 38 | Page 134
$\int x^2 \sin^{- 1} x\ dx$
Exercise 19.25 | Q 39 | Page 134
$\int\frac{\sin^{- 1} x}{x^2} \text{ dx }$
Exercise 19.25 | Q 40 | Page 134
$\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }$
Exercise 19.25 | Q 41 | Page 134
$\int \cos^{- 1} \left( 4 x^3 - 3x \right) \text{ dx }$
Exercise 19.25 | Q 42 | Page 134
$\int \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) \text{ dx }$
Exercise 19.25 | Q 43 | Page 134
$\int \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) \text{ dx }$
Exercise 19.25 | Q 44 | Page 134
$\int\left( x + 1 \right) \text{ log x dx }$
Exercise 19.25 | Q 45 | Page 134
$\int x^2 \tan^{- 1} x\text{ dx }$
Exercise 19.25 | Q 46 | Page 134

$\int\left( e^\text{log x} + \sin x \right) \text{ cos x dx }$

Exercise 19.25 | Q 47 | Page 134
$\int\frac{\left( x \tan^{- 1} x \right)}{\left( 1 + x^2 \right)^{3/2}} \text{ dx }$
Exercise 19.25 | Q 48 | Page 134
$\int \tan^{- 1} \left( \sqrt{x} \right) \text{dx }$
Exercise 19.25 | Q 49 | Page 134
$\int x^3 \tan^{- 1}\text{ x dx }$
Exercise 19.25 | Q 50 | Page 134
$\int x \sin x \cos 2x\ dx$
Exercise 19.25 | Q 51 | Page 134
$\int\left( \tan^{- 1} x^2 \right) x\ dx$
Exercise 19.25 | Q 52 | Page 134
$\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}} dx$
Exercise 19.25 | Q 53 | Page 134
$\int \sin^3 \sqrt{x}\ dx$
Exercise 19.25 | Q 54 | Page 134
$\int x \sin^3 x\ dx$
Exercise 19.25 | Q 55 | Page 134
$\int \cos^3 \sqrt{x}\ dx$
Exercise 19.25 | Q 56 | Page 134
$\int x \cos^3 x\ dx$
Exercise 19.25 | Q 57 | Page 134
$\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} dx$
Exercise 19.25 | Q 58 | Page 134
$\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx }$
Exercise 19.25 | Q 59 | Page 134
$\int\frac{x^3 \sin^{- 1} x^2}{\sqrt{1 - x^4}} \text{ dx }$
Exercise 19.25 | Q 60 | Page 134
$\int\frac{x^2 \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx }$
Exercise 19.26 [Pages 14 - 143]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.26 [Pages 14 - 143]

Exercise 19.26 | Q 1 | Page 143
$\int e^x \left( \cos x - \sin x \right) dx$
Exercise 19.26 | Q 2 | Page 143
$\int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx$
Exercise 19.26 | Q 3 | Page 143
$\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx$
Exercise 19.26 | Q 4 | Page 143
$\int e^x \left( \cot x - {cosec}^2 x \right) dx$
Exercise 19.26 | Q 5 | Page 143
$\int e^x \left( \frac{x - 1}{2 x^2} \right) dx$
Exercise 19.26 | Q 6 | Page 143
$\int e^x \sec x \left( 1 + \tan x \right) dx$
Exercise 19.26 | Q 7 | Page 143
$\int e^x \left( \tan x - \log \cos x \right) dx$
Exercise 19.26 | Q 8 | Page 143
$\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx$
Exercise 19.26 | Q 9 | Page 143
$\int e^x \left( \cot x + \log \sin x \right) dx$
Exercise 19.26 | Q 10 | Page 143
$\int e^x \frac{x - 1}{\left( x + 1 \right)^3} \text{ dx }$
Exercise 19.26 | Q 11 | Page 143
$\int e^x \left( \frac{\sin 4x - 4}{1 - \cos 4x} \right) dx$
Exercise 19.26 | Q 12 | Page 14
$\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }$
Exercise 19.26 | Q 13 | Page 143
$\int e^x \frac{1 + x}{\left( 2 + x \right)^2} \text{ dx }$
Exercise 19.26 | Q 14 | Page 143
$\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx }$
Exercise 19.26 | Q 15 | Page 143
$\int e^x \left( \log x + \frac{1}{x} \right) dx$
Exercise 19.26 | Q 16 | Page 143
$\int e^x \left( \log x + \frac{1}{x^2} \right) dx$
Exercise 19.26 | Q 17 | Page 143
$\int\frac{e^x}{x}\left\{ \text{ x }\left( \log x \right)^2 + 2 \log x \right\} dx$
Exercise 19.26 | Q 18 | Page 143
$\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }$
Exercise 19.26 | Q 19 | Page 143
∴$\int e^{2x} \left( - \sin x + 2 \cos x \right) dx$
Exercise 19.26 | Q 20 | Page 143
$\int\left( \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right) dx$
Exercise 19.26 | Q 21 | Page 143
$\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx$
Exercise 19.26 | Q 22 | Page 143
$\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx$
Exercise 19.26 | Q 23 | Page 143
$\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }$
Exercise 19.26 | Q 24 | Page 143

Evaluate the following integrals:

$\int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx$
Exercise 19.27 [Page 149]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsExercise 19.27 [Page 149]

Exercise 19.27 | Q 1 | Page 149
$\int e^{ax} \cos\ bx\ dx$
Exercise 19.27 | Q 2 | Page 149
$\int e^{ax} \text{ sin} \left( bx + C \right) dx$
Exercise 19.27 | Q 3 | Page 149
$\int\text{ cos }\left( \text{ log x } \right) \text{ dx }$
Exercise 19.27 | Q 4 | Page 149
$\int e^{2x} \cos \left( 3x + 4 \right) \text{ dx }$
Exercise 19.27 | Q 5 | Page 149
$\int e^{2x} \text{ sin x cos x dx }$
Exercise 19.27 | Q 6 | Page 149
$\int e^{2x} \sin x\ dx$
Exercise 19.27 | Q 7 | Page 149

Evaluate the following integrals:

$\int e^{2x} \text{ sin }\left( 3x + 1 \right) \text{ dx }$
Exercise 19.27 | Q 8 | Page 149
$\int e^x \sin^2 x\ dx$
Exercise 19.27 | Q 9 | Page 149
$\int\frac{1}{x^3}\text{ sin } \left( \text{ log x }\right) dx$
Exercise 19.27 | Q 10 | Page 149
$\int e^{2x} \cos^2 x\ dx$
Exercise 19.27 | Q 11 | Page 149
$\int e^{- 2x} \sin x\ dx$
Exercise 19.27 | Q 12 | Page 149
$\int x^2 e^{x^3} \cos x^3 dx$
[Pages 154 - 155]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite Integrals [Pages 154 - 155]

Q 1 | Page 154
$\int\sqrt{3 + 2x - x^2} \text{ dx}$
Q 2 | Page 154
$\int\sqrt{x^2 + x + 1} \text{ dx}$
Q 3 | Page 154
$\int\sqrt{x - x^2} dx$
Q 4 | Page 154
$\int\sqrt{1 + x - 2 x^2} \text{ dx }$
Q 5 | Page 154
$\int\cos x \sqrt{4 - \sin^2 x}\text{ dx}$
Q 6 | Page 154
$\int e^x \sqrt{e^{2x} + 1} \text{ dx}$
Q 7 | Page 154
$\int\sqrt{9 - x^2}\text{ dx}$
Q 8 | Page 154
$\int\sqrt{16 x^2 + 25} \text{ dx}$
Q 9 | Page 154
$\int\sqrt{4 x^2 - 5}\text{ dx}$
Q 10 | Page 154
$\int\sqrt{2 x^2 + 3x + 4} \text{ dx}$
Q 11 | Page 154
$\int\sqrt{3 - 2x - 2 x^2} \text{ dx}$
Q 12 | Page 154
$\int x\sqrt{x^4 + 1} \text{ dx}$
Q 13 | Page 154
$\int x^2 \sqrt{a^6 - x^6} \text{ dx}$
Q 14 | Page 154
$\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}$
Q 15 | Page 155
$\int\sqrt{2ax - x^2} \text{ dx}$
Q 16 | Page 155
$\int\sqrt{3 - x^2} \text{ dx}$
Q 17 | Page 155
$\int\sqrt{x^2 - 2x} \text{ dx}$
Q 18 | Page 155
$\int\sqrt{2x - x^2} \text{ dx}$
[Pages 158 - 159]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite Integrals [Pages 158 - 159]

Q 1 | Page 158
$\int\left( x + 1 \right) \sqrt{x^2 - x + 1} \text{ dx}$
Q 2 | Page 158
$\int\left( x + 1 \right) \sqrt{2 x^2 + 3} \text{ dx}$
Q 3 | Page 159
$\int\left( 2x - 5 \right) \sqrt{2 + 3x - x^2} \text{ dx }$
Q 4 | Page 159
$\int\left( x + 2 \right) \sqrt{x^2 + x + 1} \text{ dx }$
Q 5 | Page 159
$\int\left( 4x + 1 \right) \sqrt{x^2 - x - 2} \text{ dx }$
Q 6 | Page 159
$\int\left( x - 2 \right) \sqrt{2 x^2 - 6x + 5} \text{ dx }$
Q 7 | Page 159
$\int\left( x + 1 \right) \sqrt{x^2 + x + 1} \text{ dx }$
Q 8 | Page 159
$\int\left( 2x + 3 \right) \sqrt{x^2 + 4x + 3} \text{ dx }$
Q 9 | Page 159
$\int\left( 2x - 5 \right) \sqrt{x^2 - 4x + 3} \text{ dx }$

Q 10 | Page 159
$\int x\sqrt{x^2 + x} \text{ dx }$
Q 11 | Page 159
$\int\left( x - 3 \right)\sqrt{x^2 + 3x - 18} \text{ dx }$
Q 12 | Page 159

Evaluate the following integrals:

$\int\left( x + 3 \right)\sqrt{3 - 4x - x^2} \text{ dx }$
Q 13 | Page 159
$\int(3x + 1) \sqrt{4 - 3x - 2 x^2} \text{ dx }$
Q 14 | Page 159
$\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx$
[Pages 17 - 178]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite Integrals [Pages 17 - 178]

Q 1 | Page 176
$\int\frac{2x + 1}{\left( x + 1 \right) \left( x - 2 \right)} dx$
Q 2 | Page 176
$\int\frac{1}{x\left( x - 2 \right) \left( x - 4 \right)} dx$
Q 3 | Page 176
$\int\frac{x^2 + x - 1}{x^2 + x - 6} dx$
Q 4 | Page 176
$\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx$
Q 5 | Page 176
$\int\frac{x^2 + 1}{x^2 - 1} dx$
Q 6 | Page 176
$\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx$
Q 7 | Page 176
$\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx$
Q 8 | Page 176
$\int\frac{x^2 + 1}{x\left( x^2 - 1 \right)} dx$
Q 9 | Page 176
$\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx$
Q 10 | Page 176
$\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx$
Q 11 | Page 176
$\int\frac{\sin 2x}{\left( 1 + \sin x \right) \left( 2 + \sin x \right)} dx$
Q 12 | Page 176
$\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 3 \right)} dx$
Q 13 | Page 176
$\int\frac{1}{x \log x \left( 2 + \log x \right)} dx$
Q 14 | Page 176

Evaluate the following integral :-

$\int\frac{x^2 + x + 1}{\left( x^2 + 1 \right)\left( x + 2 \right)}dx$
Q 15 | Page 176
$\int\frac{a x^2 + bx + c}{\left( x - a \right) \left( x - b \right) \left( x - c \right)} dx,\text{ where a, b, c are distinct}$
Q 16 | Page 176

Evaluate the following integral :-

$\int\frac{x}{\left( x^2 + 1 \right)\left( x - 1 \right)}dx$
Q 17 | Page 176
$\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx$
Q 18 | Page 176

Evaluate the following integral:

$\int\frac{x^2}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)}dx$
Q 19 | Page 176
$\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx$
Q 20 | Page 176
$\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx$
Q 21 | Page 176
$\int\frac{x^2 + 1}{\left( 2x + 1 \right) \left( x^2 - 1 \right)} dx$
Q 22 | Page 177
$\int\frac{1}{x\left[ 6 \left( \log x \right)^2 + 7 \log x + 2 \right]} dx$
Q 23 | Page 177
$\int\frac{1}{x\left( x^n + 1 \right)} dx$
Q 24 | Page 177
$\int\frac{x}{\left( x^2 - a^2 \right) \left( x^2 - b^2 \right)} dx$
Q 25 | Page 177

Evaluate the following integral:

$\int\frac{x^2 + 1}{\left( x^2 + 4 \right)\left( x^2 + 25 \right)}dx$
Q 26 | Page 177

Evaluate the following integral:

$\int\frac{x^3 + x + 1}{x^2 - 1}dx$ $\int\frac{x^3 + x + 1}{x^2 - 1}dx$

Q 27 | Page 177

Evaluate the following integral:

$\int\frac{3x - 2}{\left( x + 1 \right)^2 \left( x + 3 \right)}dx$
Q 28 | Page 177
$\int\frac{2x + 1}{\left( x + 2 \right) \left( x - 3 \right)^2} dx$
Q 29 | Page 177
$\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx$
Q 30 | Page 177
$\int\frac{x}{\left( x - 1 \right)^2 \left( x + 2 \right)} dx$
Q 31 | Page 177
$\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx$
Q 32 | Page 177
$\int\frac{x^2 + x - 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx$
Q 33 | Page 177
$\int\frac{2 x^2 + 7x - 3}{x^2 \left( 2x + 1 \right)} dx$
Q 34 | Page 177
$\int\frac{5 x^2 + 20x + 6}{x^3 + 2 x^2 + x} dx$
Q 35 | Page 177
$\int\frac{18}{\left( x + 2 \right) \left( x^2 + 4 \right)} dx$
Q 36 | Page 177
$\int\frac{5}{\left( x^2 + 1 \right) \left( x + 2 \right)} dx$
Q 37 | Page 177
$\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx$
Q 38 | Page 177
$\int\frac{1}{1 + x + x^2 + x^3} dx$
Q 39 | Page 177
$\int\frac{1}{\left( x + 1 \right)^2 \left( x^2 + 1 \right)} dx$
Q 40 | Page 177
$\int\frac{2x}{x^3 - 1} dx$
Q 41 | Page 177
$\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}$
Q 42 | Page 177
$\int\frac{x^2}{\left( x^2 + 1 \right) \left( 3 x^2 + 4 \right)} dx$
Q 43 | Page 177
$\int\frac{3x + 5}{x^3 - x^2 - x + 1} dx$
Q 44 | Page 177
$\int\frac{x^3 - 1}{x^3 + x} dx$
Q 45 | Page 177
$\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx$
Q 46 | Page 177
$\int\frac{1}{x \left( x^4 + 1 \right)} dx$
Q 47 | Page 177

Evaluate the following integral:

$\int\frac{1}{x\left( x^3 + 8 \right)}dx$

Q 48 | Page 177
$\int\frac{3}{\left( 1 - x \right) \left( 1 + x^2 \right)} dx$
Q 49 | Page 177
$\int\frac{\cos x}{\left( 1 - \sin x \right)^3 \left( 2 + \sin x \right)} dx$
Q 50 | Page 177

Evaluate the following integral:

$\int\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)}dx$
Q 51 | Page 177
$\int\frac{\cos x}{\left( 1 - \sin x \right) \left( 2 - \sin x \right)} dx$
Q 52 | Page 177
$\int\frac{2x + 1}{\left( x - 2 \right) \left( x - 3 \right)} dx$
Q 53 | Page 177
$\int\frac{1}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)} dx$
Q 54 | Page 177
$\int\frac{1}{x \left( x^4 - 1 \right)} dx$
Q 55 | Page 177
$\int\frac{1}{x^4 - 1} dx$
Q 56 | Page 177
Find $\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx$
Q 57 | Page 177

Evaluate the following integrals:

$\int\frac{x^2}{(x - 1) ( x^2 + 1)}dx$
Q 58 | Page 177

Evaluate the following integral:

$\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx$
Q 59 | Page 177
$\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx$
Q 60 | Page 178
$\int\frac{1}{\sin x \left( 3 + 2 \cos x \right)} dx$
Q 61 | Page 17
$\int\frac{1}{\sin x + \sin 2x} dx$
Q 62 | Page 178
$\int\frac{x + 1}{x \left( 1 + x e^x \right)} dx$
Q 63 | Page 178
$\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx$

Q 64 | Page 178
$\int\frac{4 x^4 + 3}{\left( x^2 + 2 \right) \left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx$
Q 65 | Page 178
$\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx$
Q 66 | Page 178

Evaluate the following integral:

$\int\frac{x^2}{x^4 - x^2 - 12}dx$

Q 67 | Page 178

Evaluate the following integral:

$\int\frac{x^2}{1 - x^4}dx$
Q 68 | Page 178

Evaluate the following integral:

$\int\frac{x^2}{x^4 + x^2 - 2}dx$
Q 69 | Page 178
$\int\frac{( x^2 + 1) ( x^2 + 4)}{( x^2 + 3) ( x^2 - 5)} dx$
[Page 190]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite Integrals [Page 190]

Q 1 | Page 190
$\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }$
Q 2 | Page 190
$\int\sqrt{\cot \text{θ} d } \text{ θ}$
Q 3 | Page 190
$\int\frac{x^2 + 9}{x^4 + 81} \text{ dx }$

Q 4 | Page 190
$\int\frac{1}{x^4 + x^2 + 1} \text{ dx }$
Q 5 | Page 190
$\int\frac{x^2 - 3x + 1}{x^4 + x^2 + 1} \text{ dx }$
Q 6 | Page 190
$\int\frac{x^2 + 1}{x^4 - x^2 + 1} \text{ dx }$
Q 7 | Page 190
$\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }$
Q 8 | Page 190
$\int\frac{x^2 + 1}{x^4 + 7 x^2 + 1} 2 \text{ dx }$
Q 9 | Page 190
$\int\frac{\left( x - 1 \right)^2}{x^4 + x^2 + 1} \text{ dx}$
Q 10 | Page 190
$\int\frac{1}{x^4 + 3 x^2 + 1} \text{ dx }$
Q 11 | Page 190

Evaluate the following integral:

$\int\frac{1}{\sin^4 x + \sin^2 x \cos^2 x + \cos^4 x}dx$
[Pages 176 - 196]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite Integrals [Pages 176 - 196]

Q 1 | Page 196
$\int\frac{1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }$
Q 2 | Page 196
$\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }$
Q 3 | Page 196
$\int\frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }$
Q 4 | Page 196
$\int\frac{x^2}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx}$
Q 5 | Page 196
$\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}$
Q 6 | Page 196
$\int\frac{1}{\left( x^2 + 1 \right) \sqrt{x}} \text{ dx }$
Q 7 | Page 196
$\int\frac{x}{\left( x^2 + 2x + 2 \right) \sqrt{x + 1}} \text{ dx}$
Q 8 | Page 196
$\int\frac{1}{\left( x - 1 \right) \sqrt{x^2 + 1}} \text{ dx }$
Q 9 | Page 196
$\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }$
Q 10 | Page 196
$\int\frac{1}{\left( x^2 - 1 \right) \sqrt{x^2 + 1}} \text{ dx }$
Q 11 | Page 176
$\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 1}} \text{ dx }$
Q 12 | Page 196
$\int\frac{1}{\left( 1 + x^2 \right) \sqrt{1 - x^2}} \text{ dx }$
Q 13 | Page 176
$\int\frac{1}{\left( 2 x^2 + 3 \right) \sqrt{x^2 - 4}} \text{ dx }$
Q 14 | Page 196
$\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 9}} \text{ dx}$
Very Short Answers [Pages 179 - 198]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsVery Short Answers [Pages 179 - 198]

Very Short Answers | Q 1 | Page 197

Write a value of

$\int\frac{1 + \cot x}{x + \log \sin x} \text{ dx }$
Very Short Answers | Q 2 | Page 197

Write a value of

$\int e^{3 \text{ log x}} x^4\text{ dx}$
Very Short Answers | Q 3 | Page 197

Write a value of

$\int x^2 \sin x^3 \text{ dx }$
Very Short Answers | Q 4 | Page 197

Write a value of

$\int \tan^3 x \sec^2 x \text{ dx }$.

Very Short Answers | Q 5 | Page 197

Write a value of

$\int e^x \left( \sin x + \cos x \right) \text{ dx}$

Very Short Answers | Q 6 | Page 197

Write a value of

$\int \tan^6 x \sec^2 x \text{ dx }$ .
Very Short Answers | Q 7 | Page 197

Write a value of

$\int\frac{\cos x}{3 + 2 \sin x}\text{ dx}$
Very Short Answers | Q 8 | Page 197

Write a value of

$\int e^x \sec x \left( 1 + \tan x \right) \text{ dx }$
Very Short Answers | Q 9 | Page 197

Write a value of

$\int\frac{\log x^n}{x} \text{ dx}$
Very Short Answers | Q 10 | Page 197

Write a value of

$\int\frac{\left( \log x \right)^n}{x} \text{ dx }$
Very Short Answers | Q 11 | Page 197

Write a value of

$\int e^{\text{ log sin x }}\text{ cos x}. \text{ dx}$
Very Short Answers | Q 12 | Page 197
Write a valoue of $\int \sin^3 x \cos x\ dx$

Very Short Answers | Q 13 | Page 197

Write a value of$\int \cos^4 x \text{ sin x dx }$

Very Short Answers | Q 14 | Page 197

Write a value of$\int\text{ tan x }\sec^3 x\ dx$

Very Short Answers | Q 15 | Page 179

Write a value of$\int\frac{1}{1 + e^x} \text{ dx }$

Very Short Answers | Q 16 | Page 197

Write a value of$\int\frac{1}{1 + 2 e^x} \text{ dx }$.

Very Short Answers | Q 17 | Page 197

Write a value of$\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx$

Very Short Answers | Q 18 | Page 197

Write a value of$\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx$

Very Short Answers | Q 19 | Page 197

Write a value of$\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx$

Very Short Answers | Q 20 | Page 197

Write a value of$\int \log_e x\ dx$.

Very Short Answers | Q 21 | Page 197

Write a value of$\int a^x e^x \text{ dx }$

Very Short Answers | Q 22 | Page 197

Write a value of

$\int e^{2 x^2 + \ln x} \text{ dx}$
Very Short Answers | Q 23 | Page 197

Write a value of$\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx$ .

Very Short Answers | Q 24 | Page 197
Write a value of$\int\frac{\cos x}{\sin x \log \sin x} dx$

Very Short Answers | Q 25 | Page 197

Write a value of$\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }$

Very Short Answers | Q 26 | Page 197

Write a value of

$\int\frac{a^x}{3 + a^x} \text{ dx}$
Very Short Answers | Q 27 | Page 197

Write a value of

$\int\frac{1 + \log x}{3 + x \log x} \text{ dx }$ .
Very Short Answers | Q 28 | Page 197

Write a value of$\int\frac{\sin x}{\cos^3 x} \text{ dx }$

Very Short Answers | Q 29 | Page 197

Write a value of$\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}$

Very Short Answers | Q 30 | Page 197

Write a value of$\int\frac{1}{x \left( \log x \right)^n} \text { dx }$.

Very Short Answers | Q 31 | Page 197

Write a value of$\int e^{ax} \sin\ bx\ dx$

Very Short Answers | Q 32 | Page 197
Write a value of$\int e^{ax} \cos\ bx\ dx$.

Very Short Answers | Q 33 | Page 198

Write a value of$\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx$ .

Very Short Answers | Q 34 | Page 198

Write a value of$\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx$ .

Very Short Answers | Q 35 | Page 198

Write a value of$\int\sqrt{4 - x^2} \text{ dx }$

Very Short Answers | Q 36 | Page 198

Write a value of$\int\sqrt{9 + x^2} \text{ dx }$.

Very Short Answers | Q 37 | Page 198

Write a value of$\int\sqrt{x^2 - 9} \text{ dx}$

Very Short Answers | Q 38 | Page 198

Evaluate:$\int\frac{x^2}{1 + x^3} \text{ dx }$ .

Very Short Answers | Q 39 | Page 198

Evaluate:

$\int\frac{x^2 + 4x}{x^3 + 6 x^2 + 5} \text{ dx }$
Very Short Answers | Q 40 | Page 198

Evaluate:$\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} \text{ dx }$

Very Short Answers | Q 41 | Page 198

Evaluate:$\int\frac{\sin \sqrt{x}}{\sqrt{x}} \text{ dx }$

Very Short Answers | Q 42 | Page 198

Evaluate:$\int\frac{\cos \sqrt{x}}{\sqrt{x}} \text{ dx }$

Very Short Answers | Q 43 | Page 198

Evaluate:$\int\frac{\left( 1 + \log x \right)^2}{x} \text{ dx }$

Very Short Answers | Q 44 | Page 198

Evaluate:$\int \sec^2 \left( 7 - 4x \right) \text{ dx }$

Very Short Answers | Q 45 | Page 198

Evaluate:$\int\frac{\log x}{x} \text{ dx }$

Very Short Answers | Q 46 | Page 198

Evaluate:  $\int 2^x \text{ dx }$

Very Short Answers | Q 47 | Page 198

Write a value of $\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }$

Very Short Answers | Q 48 | Page 198

Evaluate:  $\int\frac{x^3 - 1}{x^2} \text{ dx}$

Very Short Answers | Q 49 | Page 198

Evaluate: $\int\frac{x^3 - x^2 + x - 1}{x - 1} \text{ dx }$

Very Short Answers | Q 50 | Page 198

Evaluate:$\int\frac{e\tan^{- 1} x}{1 + x^2} \text{ dx }$

Very Short Answers | Q 51 | Page 198

Evaluate: $\int\frac{1}{\sqrt{1 - x^2}} \text{ dx }$

Very Short Answers | Q 52 | Page 198

Write the value of$\int\sec x \left( \sec x + \tan x \right)\text{ dx }$

Very Short Answers | Q 53 | Page 198

Evaluate: $\int\frac{1}{x^2 + 16}\text{ dx }$

Very Short Answers | Q 54 | Page 198

Evaluate: $\int\left( 1 - x \right)\sqrt{x}\text{ dx }$

Very Short Answers | Q 55 | Page 198

Evaluate: $\int\frac{x + \cos6x}{3 x^2 + \sin6x}\text{ dx }$

Very Short Answers | Q 56 | Page 198
$\text{ If } \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, \text{ then write the value of f}\left( x \right) .$
Very Short Answers | Q 57 | Page 198
$If \int e^x \left( \tan x + 1 \right)\text{ sec x dx } = e^x f\left( x \right) + C, \text{ then write the value of f}\left( x \right) .$

Very Short Answers | Q 58 | Page 198

Evaluate:  $\int\frac{2}{1 - \cos2x}\text{ dx }$

Very Short Answers | Q 59 | Page 198

Write the anti-derivative of  $\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .$

Very Short Answers | Q 60 | Page 198

Evaluate:  $\int \cos^{- 1} \left( \sin x \right) \text{ dx }$

Very Short Answers | Q 61 | Page 198

Evaluate:

$\int\frac{1}{\sin^2 x \cos^2 x}dx$
Very Short Answers | Q 62 | Page 198

Evaluate : $\int\frac{1}{x(1 + \log x)} \text{ dx}$

MCQ [Pages 199 - 203]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsMCQ [Pages 199 - 203]

MCQ | Q 1 | Page 199
$\int\frac{x}{4 + x^4} \text{ dx }$ is equal to
• $\frac{1}{4} \tan^{- 1} x^2 + C$

• $\frac{1}{4} \tan^{- 1} \left( \frac{x^2}{2} \right)$

• $\frac{1}{2} \tan^{- 1} \left( \frac{x^2}{2} \right)$

• none of these

MCQ | Q 2 | Page 199
$\int\frac{1}{\cos x + \sqrt{3} \sin x} \text{ dx }$ is equal to
•   log   tan (x/3  + π / 2) + C 

• $\text{ log tan} \left( \frac{x}{2} - \frac{\pi}{3} \right) + C$

•    1/2  log   tan (x/2  + π /3 ) + C 

• none of these

MCQ | Q 3 | Page 200
\int \text{ x} \text{ sec x}^2 \text{  dx  is  equal  to }

• $\frac{1}{2}$ log (sec x2 + tan x2) + C

• $\frac{x^2}{2}$  log (sec x2 + tan x2) + C

• 2 log (sec x2 + tan x2) + C

• none of these

MCQ | Q 4 | Page 200

If $\int\frac{1}{5 + 4 \sin x} dx = A \tan^{- 1} \left( B \tan\frac{x}{2} + \frac{4}{3} \right) + C,$ then

•  A =$\frac{2}{3}$, B =$\frac{5}{3}$

•  A =$\frac{1}{3}$, B = $\frac{2}{3}$

•  A =$- \frac{2}{3}$, B =$\frac{5}{3}$

• A =$\frac{1}{3}$, B =$- \frac{5}{3}$

MCQ | Q 5 | Page 200
$\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx$ is equal to
•  xsin x + C

•  xsin x cos x + C

• $\frac{\left( x^{\sin x} \right)^2}{2} + C$

• none of these

MCQ | Q 6 | Page 200

Integration of $\frac{1}{1 + \left( \log_e x \right)^2}$ with respect to loge x is

• $\frac{\tan^{- 1} \left( \log_e x \right)}{x} + C$

• $\tan^{- 1} \left( \log_e x \right) + C$

• $\frac{\tan^{- 1} x}{x} + C$

• none of these

MCQ | Q 7 | Page 200

If $\int\frac{\cos 8x + 1}{\tan 2x - \cot 2x} dx$

• $- \frac{1}{16}$

• $\frac{1}{8}$

• $\frac{1}{16}$

• $- \frac{1}{8}$

MCQ | Q 8 | Page 200

If $\int\frac{\sin^8 x - \cos^8 x}{1 - 2 \sin^2 x \cos^2 x} dx$

• -1/2

• 1/2

• -1

• 1

MCQ | Q 9 | Page 200
$\int\left( x - 1 \right) e^{- x} dx$ is equal to
• − xex + C

• xex + C

• − xex + C

• xex + C

MCQ | Q 10 | Page 200

If $\int\frac{2^{1/x}}{x^2} dx = k 2^{1/x} + C,$  then k is equal to

• $- \frac{1}{\log_e 2}$

• − loge 2

• -1`

• $\frac{1}{2}$

MCQ | Q 11 | Page 200
$\int\frac{1}{1 + \tan x} dx =$
• loge (x + sin x) + C

• loge (sin x + cos x) + C

• $2 \sec^2 \frac{x}{2} + C$

• $\frac{1}{2}$ [x + log (sin x + cos x)] + C

MCQ | Q 12 | Page 200
$\int \left| x \right|^3 dx$ is equal to
• $\frac{- x^4}{4} + C$

• $\frac{\left| x \right|^4}{4} + C$

• $\frac{x^4}{4} + C$

• none of these

MCQ | Q 13 | Page 200

The value of $\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx$ is

• 2 cos $\sqrt{x}$

• $\sqrt{\frac{\cos x}{x}} + C$

• sin $\sqrt{x} + C$

• 2 sin $\sqrt{x} + C$

MCQ | Q 14 | Page 201
$\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =$
•  ex cot x + C

• ex cot x + C

•  ex cosec x + C

• ex cosec x + C

MCQ | Q 15 | Page 201
$\int\frac{\sin^6 x}{\cos^8 x} dx =$
•  tan 7x + C

• $\frac{\tan^7 x}{7} + C$
• $\frac{\tan 7x}{7} + C$
• sec7 x + C

MCQ | Q 16 | Page 201
$\int\frac{1}{7 + 5 \cos x} dx =$
• $\frac{1}{\sqrt{6}} \tan^{- 1} \left( \frac{1}{\sqrt{6}}\tan\frac{x}{2} \right) + C$
• $\frac{1}{\sqrt{3}} \tan^{- 1} \left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + C$

• $\frac{1}{4} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C$
• $\frac{1}{7} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C$
MCQ | Q 17 | Page 201
$\int\frac{1}{1 - \cos x - \sin x} dx =$
• $\log\left| 1 + \cot\frac{x}{2} \right| + C$
• $\log\left| 1 - \tan\frac{x}{2} \right| + C$
• $\log\left| 1 - \cot\frac{x}{2} \right| + C$
• $\log\left| 1 + \tan\frac{x}{2} \right| + C$
MCQ | Q 18 | Page 201

$\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =$

• $\frac{e^x}{x + 4} + C$

• $\frac{e^x}{x + 3} + C$

• $\frac{1}{\left( x + 4 \right)^2} + C$

• $\frac{e^x}{\left( x + 4 \right)^2} + C$

MCQ | Q 19 | Page 201
$\int\frac{\sin x}{3 + 4 \cos^2 x} dx$
• log (3 + 4 cos2 x) + C

• $\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{\cos x}{\sqrt{3}} \right) + C$
• $- \frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C$
• $\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C$
MCQ | Q 20 | Page 201
$\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx$
• $- e^x \tan\frac{x}{2} + C$
• $- e^x \cot\frac{x}{2} + C$
• $- \frac{1}{2} e^x \tan\frac{x}{2} + C$
• $- \frac{1}{2} e^x \cot\frac{x}{2} + C$
MCQ | Q 21 | Page 201
$\int\frac{2}{\left( e^x + e^{- x} \right)^2} dx$
• $\frac{- e^{- x}}{e^x + e^{- x}} + C$
• $- \frac{1}{e^x + e^{- x}} + C$
• $\frac{- 1}{\left( e^x + 1 \right)^2} + C$
• $\frac{1}{e^x - e^{- x}} + C$
MCQ | Q 22 | Page 201
$\int\frac{e^x \left( 1 + x \right)}{\cos^2 \left( x e^x \right)} dx =$
• 2 loge cos (xex) + C

• sec (xex) + C

• tan (xex) + C

•  tan (x + ex) + C

MCQ | Q 23 | Page 201
$\int\frac{\sin^2 x}{\cos^4 x} dx =$
• $\frac{1}{3} \tan^2 x + C$
• $\frac{1}{2} \tan^2 x + C$
• $\frac{1}{3} \tan^3 x + C$
• none of these

MCQ | Q 24 | Page 202

The primitive of the function $f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}$

• $\frac{a^{x + \frac{1}{x}}}{\log_e a}$
• $\log_e a \cdot a^{x + \frac{1}{x}}$
• $\frac{a^{x + \frac{1}{x}}}{x} \log_e a$
• $x\frac{a^{x + \frac{1}{x}}}{\log_e a}$
MCQ | Q 25 | Page 202

The value of $\int\frac{1}{x + x \log x} dx$ is

• 1 + log x

• x + log x

• x log (1 + log x)

• log (1 + log x)

MCQ | Q 26 | Page 202

$\int\sqrt{\frac{x}{1 - x}} dx$  is equal to

• $\sin^{- 1} \sqrt{x} + C$
• $\sin^{- 1} \left\{ \sqrt{x} - \sqrt{x \left( 1 - x \right)} \right\} + C$
• $\sin^{- 1} \left\{ \sqrt{x \left( 1 - x \right)} \right\} + C$
• $\sin^{- 1} \sqrt{x} - \sqrt{x \left( 1 - x \right)} + C$
MCQ | Q 27 | Page 202
$\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx =$

• ex f (x) + C

• ex + (x)

•  2ex f (x)

•  ex − f (x)

MCQ | Q 28 | Page 202

The value of $\int\frac{\sin x + \cos x}{\sqrt{1 - \sin 2x}} dx$ is equal to

• $\sqrt{\sin 2x} + C$
• $\sqrt{\cos 2x} + C$
•  ± (sin x − cos x) + C

•  ± log (sin x − cos x) + C

MCQ | Q 30 | Page 202
$\int\frac{\cos 2x - 1}{\cos 2x + 1} dx =$
• tan x − x + C

• x + tan x + C

• x − tan x + C

• − x − cot x + C

MCQ | Q 31 | Page 202
$\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx$ is equal to
• $2\left( \sin x + x\cos\theta \right) + C$

• $2\left( \sin x - x\cos\theta \right) + C$
• $2\left( \sin x + 2x\cos\theta \right) + C$

• $2\left( \sin x - 2x\cos\theta \right) + C$
MCQ | Q 32 | Page 202
$\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx$  is equal to
• $\frac{1}{5x} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C$

• $\frac{1}{5} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C$

• $\frac{1}{10x} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C$

• $\frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C$

MCQ | Q 33 | Page 202

$\int\frac{x^3}{\sqrt{1 + x^2}}dx = a \left( 1 + x^2 \right)^\frac{3}{2} + b\sqrt{1 + x^2} + C$, then

• $a = \frac{1}{3}, b = 1$

• $a = - \frac{1}{3}, b = 1$

• $a = - \frac{1}{3}, b = - 1$

• $a = \frac{1}{3}, b = - 1$

MCQ | Q 34 | Page 202
$\int\frac{x^3}{x + 1}dx$ is equal to
• $x + \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 - x \right| + C$

• $x + \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 - x \right| + C$

• $x - \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 + x \right| + C$

• $x - \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 + x \right| + C$

MCQ | Q 35 | Page 203

If $\int\frac{1}{\left( x + 2 \right)\left( x^2 + 1 \right)}dx = a\log\left| 1 + x^2 \right| + b \tan^{- 1} x + \frac{1}{5}\log\left| x + 2 \right| + C,$ then

• $a = - \frac{1}{10}, b = - \frac{2}{5}$

• $a = \frac{1}{10}, b = - \frac{2}{5}$

• $a = - \frac{1}{10}, b = \frac{2}{5}$

• $a = \frac{1}{10}, b = \frac{2}{5}$
Revision Excercise [Pages 203 - 205]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsRevision Excercise [Pages 203 - 205]

Revision Excercise | Q 1 | Page 203

$\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }$

Revision Excercise | Q 2 | Page 203

$\int\frac{1 - x^4}{1 - x} \text{ dx }$

Revision Excercise | Q 3 | Page 203

$\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }$

Revision Excercise | Q 4 | Page 203

$\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }$

Revision Excercise | Q 5 | Page 203

$\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}$

Revision Excercise | Q 6 | Page 203
$\int\frac{\left( 2^x + 3^x \right)^2}{6^x} \text{ dx }$
Revision Excercise | Q 7 | Page 203
$\int\frac{\sin x}{1 + \sin x} \text{ dx }$
Revision Excercise | Q 8 | Page 203
$\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}$
Revision Excercise | Q 9 | Page 203
$\int \sec^2 x \cos^2 2x \text{ dx }$
Revision Excercise | Q 10 | Page 203
$\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx}$
Revision Excercise | Q 11 | Page 203
$\int \sin^4 2x\ dx$
Revision Excercise | Q 12 | Page 203
$\int \cos^3 (3x)\ dx$
Revision Excercise | Q 13 | Page 203
$\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}$
Revision Excercise | Q 14 | Page 203
$\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx}$
Revision Excercise | Q 15 | Page 203
$\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }$
Revision Excercise | Q 16 | Page 203
$\int\frac{1}{e^x + 1} \text{ dx }$
Revision Excercise | Q 17 | Page 203
$\int\frac{e^x - 1}{e^x + 1} \text{ dx}$
Revision Excercise | Q 18 | Page 203
$\int\frac{1}{e^x + e^{- x}} dx$
Revision Excercise | Q 19 | Page 203
$\int\frac{\cos^7 x}{\sin x} dx$
Revision Excercise | Q 20 | Page 203

$\int\sin x \sin 2x \text{ sin 3x dx }$

Revision Excercise | Q 21 | Page 203

$\int\text{ cos x cos 2x cos 3x dx}$

Revision Excercise | Q 22 | Page 203
$\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \text{ dx}$
Revision Excercise | Q 23 | Page 203
$\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }$

Revision Excercise | Q 24 | Page 203
$\int\frac{1}{\text{ sin} \left( x - a \right) \text{ sin } \left( x - b \right)} \text{ dx }$
Revision Excercise | Q 25 | Page 203
$\int\frac{1}{\text{ cos }\left( x - a \right) \text{ cos }\left( x - b \right)} \text{ dx }$
Revision Excercise | Q 26 | Page 203
$\int\frac{\sin x}{\sqrt{1 + \sin x}} dx$
Revision Excercise | Q 27 | Page 203
$\int\frac{\sin x}{\cos 2x} \text{ dx }$
Revision Excercise | Q 28 | Page 203
$\int \tan^3 x\ dx$
Revision Excercise | Q 29 | Page 203
$\int \tan^4 x\ dx$
Revision Excercise | Q 30 | Page 203
$\int \tan^5 x\ dx$
Revision Excercise | Q 31 | Page 203
$\int \cot^4 x\ dx$
Revision Excercise | Q 32 | Page 203
$\int \cot^5 x\ dx$
Revision Excercise | Q 33 | Page 203
$\int\frac{x^2}{\left( x - 1 \right)^3} dx$
Revision Excercise | Q 34 | Page 203
$\int x\sqrt{2x + 3} \text{ dx }$
Revision Excercise | Q 35 | Page 203
$\int\frac{x^3}{\left( 1 + x^2 \right)^2} \text{ dx }$
Revision Excercise | Q 36 | Page 203
$\int x \sin^5 x^2 \cos x^2 dx$
Revision Excercise | Q 38 | Page 203
$\int \sin^5 x\ dx$
Revision Excercise | Q 39 | Page 203
$\int \cos^5 x\ dx$
Revision Excercise | Q 40 | Page 203
$\int\sqrt{\sin x} \cos^3 x\ \text{ dx }$
Revision Excercise | Q 41 | Page 203
$\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }$
Revision Excercise | Q 42 | Page 203
$\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }$
Revision Excercise | Q 43 | Page 203
$\int\frac{1}{\sqrt{x^2 + a^2}} \text{ dx }$
Revision Excercise | Q 44 | Page 203
$\int\frac{1}{4 x^2 + 4x + 5} dx$
Revision Excercise | Q 45 | Page 203
$\int\frac{1}{x^2 + 4x - 5} \text{ dx }$
Revision Excercise | Q 46 | Page 203
$\int\frac{1}{1 - x - 4 x^2}\text{ dx }$
Revision Excercise | Q 47 | Page 203
$\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }$
Revision Excercise | Q 48 | Page 203
$\int\frac{\sin x}{\sqrt{\cos^2 x - 2 \cos x - 3}} \text{ dx }$
Revision Excercise | Q 49 | Page 204
$\int\sqrt{\text{ cosec x} - 1} \text{ dx }$
Revision Excercise | Q 50 | Page 204
$\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}$
Revision Excercise | Q 51 | Page 204
$\int\frac{x + 1}{x^2 + 4x + 5} \text{ dx}$
Revision Excercise | Q 52 | Page 204
$\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }$
Revision Excercise | Q 53 | Page 204
$\int\sqrt{\frac{1 + x}{x}} \text{ dx }$
Revision Excercise | Q 54 | Page 204

$\int\sqrt{\frac{1 - x}{x}} \text{ dx}$

Revision Excercise | Q 55 | Page 204
$\int\frac{\sqrt{a} - \sqrt{x}}{1 - \sqrt{ax}}\text{ dx }$
Revision Excercise | Q 56 | Page 204
$\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }$
Revision Excercise | Q 57 | Page 204

$\int\frac{1}{4 \sin^2 x + 4 \sin x \cos x + 5 \cos^2 x} \text{ dx }$

Revision Excercise | Q 58 | Page 204
$\int\frac{1}{a + b \tan x} \text{ dx }$
Revision Excercise | Q 59 | Page 204
$\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }$
Revision Excercise | Q 60 | Page 204

$\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }$

Revision Excercise | Q 61 | Page 204

$\int\frac{x^3}{\sqrt{x^8 + 4}} \text{ dx }$

Revision Excercise | Q 62 | Page 204

$\int\frac{1}{2 - 3 \cos 2x} \text{ dx }$

Revision Excercise | Q 63 | Page 204
$\int\frac{\cos x}{\frac{1}{4} - \cos^2 x} \text{ dx }$
Revision Excercise | Q 64 | Page 204
$\int\frac{1}{1 + 2 \cos x} \text{ dx }$
Revision Excercise | Q 65 | Page 204
$\int\frac{1}{1 - 2 \sin x} \text{ dx }$
Revision Excercise | Q 66 | Page 204
$\int\frac{1}{\sin x \left( 2 + 3 \cos x \right)} \text{ dx }$
Revision Excercise | Q 67 | Page 204
$\int\frac{1}{\sin x + \sin 2x} \text{ dx }$
Revision Excercise | Q 68 | Page 204

$\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}$

Revision Excercise | Q 69 | Page 204
$\int\frac{1}{5 - 4 \sin x} \text{ dx }$
Revision Excercise | Q 70 | Page 205

$\int \sec^4 x\ dx$

Revision Excercise | Q 71 | Page 204

$\int {cosec}^4 2x\ dx$

Revision Excercise | Q 72 | Page 204

$\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }$

Revision Excercise | Q 73 | Page 204

$\int\frac{1}{2 + \cos x} \text{ dx }$

Revision Excercise | Q 74 | Page 204
$\int\sqrt{\frac{a + x}{x}}dx$

Revision Excercise | Q 75 | Page 204
$\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} \text{ dx}$
Revision Excercise | Q 76 | Page 204
$\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }$
Revision Excercise | Q 77 | Page 204
$\int\frac{\cos^5 x}{\sin x} \text{ dx }$
Revision Excercise | Q 78 | Page 204
$\int\frac{\sin^6 x}{\cos x} \text{ dx }$
Revision Excercise | Q 79 | Page 204
$\int\frac{\sin^2 x}{\cos^6 x} \text{ dx }$
Revision Excercise | Q 80 | Page 204
$\int \sec^6 x\ dx$
Revision Excercise | Q 81 | Page 204
$\int \tan^5 x\ \sec^3 x\ dx$
Revision Excercise | Q 82 | Page 204
$\int \tan^3 x\ \sec^4 x\ dx$
Revision Excercise | Q 83 | Page 204
$\int\frac{1}{\sec x + cosec x}\text{ dx }$
Revision Excercise | Q 84 | Page 204
$\int\sqrt{a^2 + x^2} \text{ dx }$
Revision Excercise | Q 85 | Page 204
$\int\sqrt{x^2 - a^2} \text{ dx}$
Revision Excercise | Q 86 | Page 204
$\int\sqrt{a^2 - x^2}\text{ dx }$
Revision Excercise | Q 87 | Page 204
$\int\sqrt{3 x^2 + 4x + 1}\text{ dx }$
Revision Excercise | Q 88 | Page 204
$\int\sqrt{1 + 2x - 3 x^2}\text{ dx }$
Revision Excercise | Q 89 | Page 204
$\int x\sqrt{1 + x - x^2}\text{ dx }$
Revision Excercise | Q 90 | Page 204
$\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}$
Revision Excercise | Q 91 | Page 204

$\int\left( 1 + x^2 \right) \ \cos 2x \ dx$

Revision Excercise | Q 92 | Page 204
$\int \log_{10} x\ dx$
Revision Excercise | Q 93 | Page 204
$\int\frac{\log \left( \log x \right)}{x} \text{ dx}$
Revision Excercise | Q 94 | Page 204
$\int x \sec^2 2x\ dx$
Revision Excercise | Q 95 | Page 204
$\int x \sin^3 x\ dx$
Revision Excercise | Q 96 | Page 204
$\int \left( x + 1 \right)^2 e^x \text{ dx }$
Revision Excercise | Q 97 | Page 204
$\int\log \left( x + \sqrt{x^2 + a^2} \right) \text{ dx}$
Revision Excercise | Q 98 | Page 204
$\int\frac{\log x}{x^3} \text{ dx }$
Revision Excercise | Q 99 | Page 204
$\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}$
Revision Excercise | Q 100 | Page 204
$\int x^3 \left( \log x \right)^2\text{ dx }$
Revision Excercise | Q 101 | Page 204
$\int\frac{1}{x \sqrt{1 + x^n}} \text{ dx}$
Revision Excercise | Q 102 | Page 204
$\int\frac{x^2}{\sqrt{1 - x}} \text{ dx }$
Revision Excercise | Q 103 | Page 204
$\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }$
Revision Excercise | Q 104 | Page 204
$\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }$
Revision Excercise | Q 105 | Page 204
$\int x\sqrt{\frac{1 - x}{1 + x}} \text{ dx }$
Revision Excercise | Q 106 | Page 204
$\int\frac{1}{x\sqrt{1 + x^3}} \text{ dx}$
Revision Excercise | Q 107 | Page 204
$\int\frac{\sin x + \cos x}{\sin^4 x + \cos^4 x} \text{ dx }$
Revision Excercise | Q 108 | Page 204
$\int x^2 \tan^{- 1} x\ dx$
Revision Excercise | Q 109 | Page 205
$\int \tan^{- 1} \sqrt{x}\ dx$
Revision Excercise | Q 110 | Page 205
$\int \sin^{- 1} \sqrt{x}\ dx$
Revision Excercise | Q 111 | Page 204
$\int \sec^{- 1} \sqrt{x}\ dx$
Revision Excercise | Q 112 | Page 204
$\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} \text{ dx }$
Revision Excercise | Q 113 | Page 205
$\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}$
Revision Excercise | Q 114 | Page 205
$\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}$
Revision Excercise | Q 115 | Page 205
$\int \left( \sin^{- 1} x \right)^3 dx$
Revision Excercise | Q 116 | Page 204
$\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }$
Revision Excercise | Q 117 | Page 205
$\int\frac{x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx}$
Revision Excercise | Q 118 | Page 205
$\int e^{2x} \left( \frac{1 + \sin 2x}{1 + \cos 2x} \right) dx$
Revision Excercise | Q 119 | Page 205
$\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx}$
Revision Excercise | Q 120 | Page 205
$\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }$
Revision Excercise | Q 121 | Page 205
$\int\frac{e^{m \tan^{- 1} x}}{\left( 1 + x^2 \right)^{3/2}} \text{ dx}$
Revision Excercise | Q 122 | Page 205
$\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}$
Revision Excercise | Q 123 | Page 205
$\int\frac{x}{x^3 - 1} \text{ dx}$
Revision Excercise | Q 124 | Page 205
$\int\frac{1}{1 + x + x^2 + x^3} \text{ dx }$
Revision Excercise | Q 125 | Page 205
$\int\frac{1}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)} \text{ dx}$
Revision Excercise | Q 126 | Page 205
$\int\frac{x^2 - 2}{x^5 - x} \text{ dx}$
Revision Excercise | Q 127 | Page 205
$\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}$
Revision Excercise | Q 128 | Page 205
$\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}$
Revision Excercise | Q 129 | Page 205
$\int\frac{\sin 4x - 2}{1 - \cos 4x} e^{2x} \text{ dx}$
Revision Excercise | Q 130 | Page 205
$\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}$
Revision Excercise [Page 203]

### RD Sharma solutions for Class 12 Maths Chapter 19 Indefinite IntegralsRevision Excercise [Page 203]

Revision Excercise | Q 37 | Page 203
$\int \sin^3 x \cos^4 x\ \text{ dx }$

## Chapter 19: Indefinite Integrals

Exercise 19.1Exercise 19.2Exercise 19.3Exercise 19.4Exercise 19.5Exercise 19.6Exercise 19.7Exercise 19.8Exercise 19.85Exercise 19.9Exercise 19.11Exercise 19..11Exercise 19.12Exercise 19.13Exercise 19.14Exercise 19.15Exercise 19.16Exercise 19.17Exercise 19.18Exercise 19.19Exercise 19.21Exercise 19.22Exercise 19.23Exercise 19.24Exercise 19.25Exercise 19.26Exercise 19.27OthersVery Short AnswersMCQRevision Excercise

## RD Sharma solutions for Class 12 Maths chapter 19 - Indefinite Integrals

RD Sharma solutions for Class 12 Maths chapter 19 (Indefinite Integrals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 19 Indefinite Integrals are Definite Integrals Problems, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Geometrical Interpretation of Indefinite Integral, Integrals of Some Particular Functions, Indefinite Integral by Inspection, Properties of Indefinite Integral, Integration Using Trigonometric Identities, Introduction of Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Fundamental Theorem of Calculus, Definite Integral as the Limit of a Sum, Evaluation of Simple Integrals of the Following Types and Problems, Methods of Integration: Integration by Parts, Methods of Integration: Integration Using Partial Fractions, Methods of Integration: Integration by Substitution, Integration as an Inverse Process of Differentiation.

Using RD Sharma Class 12 solutions Indefinite Integrals exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 19 Indefinite Integrals Class 12 extra questions for Class 12 Maths and can use Shaalaa.com to keep it handy for your exam preparation