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

## Chapter 19: Indefinite Integrals

Ex. 19.1Ex. 19.2Ex. 19.3Ex. 19.4Ex. 19.5Ex. 19.6Ex. 19.7Ex. 19.8Ex. 19.85Ex. 19.9Ex. 19.10Ex. 19.1119..11Ex. 19.12Ex. 19.13Ex. 19.14Ex. 19.15Ex. 19.16Ex. 19.17Ex. 19.18Ex. 19.19Ex. 19.20Ex. 19.21Ex. 19.22Ex. 19.23Ex. 19.24Ex. 19.25Ex. 19.26Ex. 19.27OthersVery Short AnswersMCQRevision Excercise

#### Chapter 19: Indefinite Integrals Exercise 19.1 solutions [Page 4]

Ex. 19.1 | Q 1.1 | Page 4

Evaluate of the following integral:

(i)  $\int x^4 dx$

Ex. 19.1 | Q 1.2 | Page 4

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate of the following integral:

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

Evaluate:

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

Evaluate:

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

Evaluate :

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

Evaluate:

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

Evaluate:

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

Evaluate:

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

Evaluate:

$\int\frac{e\log \sqrt{x}}{x}dx$

#### Chapter 19: Indefinite Integrals Exercise 19.2 solutions [Pages 14 - 15]

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

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

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

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

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

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

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

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

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

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

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

Ex. 19.2 | Q 46 | Page 15

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

Ex. 19.2 | Q 47 | Page 15

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

Ex. 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)

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

#### Chapter 19: Indefinite Integrals Exercise 19.3 solutions [Pages 23 - 24]

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

 ∫  1/ {1+ cos   3x}   dx

Ex. 19.3 | Q 14 | Page 23

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

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

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

Ex. 19.3 | Q 19 | Page 24
$\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx$

#### Chapter 19: Indefinite Integrals Exercise 19.4 solutions [Page 30]

Ex. 19.4 | Q 1 | Page 30

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

Ex. 19.4 | Q 2 | Page 30
$\int\frac{x^3}{x - 2} dx$
Ex. 19.4 | Q 3 | Page 30
$\int\frac{x^2 + x + 5}{3x + 2} dx$
Ex. 19.4 | Q 4 | Page 30
$\int\frac{2x + 3}{\left( x - 1 \right)^2} dx$
Ex. 19.4 | Q 5 | Page 30
$\int\frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} dx$
Ex. 19.4 | Q 6 | Page 30
$\int\frac{2x - 1}{\left( x - 1 \right)^2} dx$

#### Chapter 19: Indefinite Integrals Exercise 19.5 solutions [Page 33]

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

#### Chapter 19: Indefinite Integrals Exercise 19.6 solutions [Pages 3 - 36]

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

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

Ex. 19.6 | Q 3 | Page 36

∫     cos ^4  2x   dx

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

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

 ∫   sin x  \sqrt (1-cos 2x)    dx

#### Chapter 19: Indefinite Integrals Exercise 19.7 solutions [Page 38]

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

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

Integrate the following integrals:

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

Integrate the following integrals:

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

#### Chapter 19: Indefinite Integrals Exercise 19.8, 19.85 solutions [Pages 47 - 48]

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

Evaluate the following integrals:

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

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

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

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

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

#### Chapter 19: Indefinite Integrals Exercise 19.9 solutions [Pages 57 - 59]

Ex. 19.9 | Q 1 | Page 57
 ∫  log x / x  dx

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

  =  ∫ root (3){ cos^2 x}  sin x   dx

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

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

Ex. 19.9 | Q 8 | Page 57

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

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

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

Ex. 19.9 | Q 11 | Page 58

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

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

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

Ex. 19.9 | Q 17 | Page 58

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

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

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

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

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

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

#### Chapter 19: Indefinite Integrals Exercise 19.10 solutions [Page 65]

Ex. 19.10 | Q 1 | Page 65
$\int x^2 \sqrt{x + 2} \text{ dx }$
Ex. 19.10 | Q 2 | Page 65
$\int\frac{x^2}{\sqrt{x - 1}} dx$
Ex. 19.10 | Q 3 | Page 65
$\int\frac{x^2}{\sqrt{3x + 4}} dx$
Ex. 19.10 | Q 4 | Page 65
$\int\frac{2x - 1}{\left( x - 1 \right)^2} dx$
Ex. 19.10 | Q 5 | Page 65
$\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }$
Ex. 19.10 | Q 6 | Page 65
$\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx$
Ex. 19.10 | Q 7 | Page 65
$\int\frac{x^2}{\sqrt{1 - x}} dx$
Ex. 19.10 | Q 8 | Page 65
$\ \int\ x \left( 1 - x \right)^{23} dx$

Ex. 19.10 | Q 9 | Page 65
$\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx$
Ex. 19.10 | Q 10 | Page 65

 ∫   1 /{x^{1/3} ( x^{1/3} -1)}    dx

#### Chapter 19: Indefinite Integrals Exercise 19.11, 19..11 solutions [Page 69]

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

 ∫   tan   x   sec^4  x   dx

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

 ∫      tan^5    x   dx

Ex. 19.11 | Q 6 | Page 69

 ∫    \sqrt{tan x}     sec^4  x   dx

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

#### Chapter 19: Indefinite Integrals Exercise 19.12 solutions [Page 73]

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

 = ∫     1/{sin^3  x cos^ 2 x }   dx

Ex. 19.12 | Q 13 | Page 73
$\int\frac{1}{\sin x \cos^3 x} dx$

#### Chapter 19: Indefinite Integrals Exercise 19.13 solutions [Page 79]

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

Evaluate the following integrals:

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

Evaluate the following integrals:

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

Evaluate the following integrals:

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

Evaluate the following integrals:

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

#### Chapter 19: Indefinite Integrals Exercise 19.14 solutions [Page 83]

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

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

#### Chapter 19: Indefinite Integrals Exercise 19.15 solutions [Page 86]

Ex. 19.15 | Q 1 | Page 86
$\int\frac{1}{4 x^2 + 12x + 5} dx$
Ex. 19.15 | Q 2 | Page 86
$\int\frac{1}{x^2 - 10x + 34} dx$
Ex. 19.15 | Q 3 | Page 86
$\int\frac{1}{1 + x - x^2} \text{ dx }$
Ex. 19.15 | Q 4 | Page 86
$\int\frac{1}{2 x^2 - x - 1} dx$
Ex. 19.15 | Q 5 | Page 86
$\int\frac{1}{x^2 + 6x + 13} dx$

#### Chapter 19: Indefinite Integrals Exercise 19.16 solutions [Page 90]

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

#### Chapter 19: Indefinite Integrals Exercise 19.17 solutions [Page 93]

Ex. 19.17 | Q 1 | Page 93
$\int\frac{1}{\sqrt{2x - x^2}} dx$
Ex. 19.17 | Q 2 | Page 93
$\int\frac{1}{\sqrt{8 + 3x - x^2}} dx$
Ex. 19.17 | Q 3 | Page 93
$\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx$
Ex. 19.17 | Q 4 | Page 93
$\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx$
Ex. 19.17 | Q 5 | Page 93
$\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)$
Ex. 19.17 | Q 6 | Page 93
$\int\frac{1}{\sqrt{7 - 3x - 2 x^2}} dx$
Ex. 19.17 | Q 7 | Page 93
$\int\frac{1}{\sqrt{16 - 6x - x^2}} dx$
Ex. 19.17 | Q 8 | Page 93
$\int\frac{1}{\sqrt{7 - 6x - x^2}} dx$
Ex. 19.17 | Q 9 | Page 93
$\int\frac{1}{\sqrt{5 x^2 - 2x}} dx$

#### Chapter 19: Indefinite Integrals Exercise 19.18 solutions [Pages 98 - 99]

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

#### Chapter 19: Indefinite Integrals Exercise 19.19 solutions [Page 104]

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

 ∫  {x-3} /{ x^2 + 2x - 4 } dx

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

Evaluate the following integrals:

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

#### Chapter 19: Indefinite Integrals Exercise 19.20 solutions [Page 106]

Ex. 19.20 | Q 1 | Page 106
$\int\frac{x^2 + x + 1}{x^2 - x} dx$
Ex. 19.20 | Q 2 | Page 106
$\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }$
Ex. 19.20 | Q 3 | Page 106
$\int\frac{\left( 1 - x^2 \right)}{x \left( 1 - 2x \right)} \text {dx$
Ex. 19.20 | Q 4 | Page 106
$\int\frac{x^2 + 1}{x^2 - 5x + 6} \text{ dx }$

Ex. 19.20 | Q 5 | Page 106
$\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }$
Ex. 19.20 | Q 5 | Page 106
$\int\frac{x^2}{x^2 + 7x + 10} dx$
Ex. 19.20 | Q 6 | Page 106
$\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }$
Ex. 19.20 | Q 7 | Page 106
$\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx$
Ex. 19.20 | Q 8 | Page 106
$\int\frac{x^3 + x^2 + 2x + 1}{x^2 - x + 1}\text{ dx }$
Ex. 19.20 | Q 9 | Page 106
$\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }$
Ex. 19.20 | Q 10 | Page 106
$\int\frac{x^2}{x^2 + 6x + 12} \text{ dx }$

#### Chapter 19: Indefinite Integrals Exercise 19.21 solutions [Pages 110 - 111]

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

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

Ex. 19.21 | Q 6 | Page 110

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

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

Evaluate the following integrals:

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

#### Chapter 19: Indefinite Integrals Exercise 19.22 solutions [Page 114]

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

#### Chapter 19: Indefinite Integrals Exercise 19.23 solutions [Page 117]

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

#### Chapter 19: Indefinite Integrals Exercise 19.24 solutions [Page 122]

Ex. 19.24 | Q 1 | Page 122
$\int\frac{1}{1 - \cot x} dx$
Ex. 19.24 | Q 2 | Page 122
$\int\frac{1}{1 - \tan x} \text{ dx }$
Ex. 19.24 | Q 3 | Page 122
$\int\frac{3 + 2 \cos x + 4 \sin x}{2 \sin x + \cos x + 3} \text{ dx }$
Ex. 19.24 | Q 4 | Page 122
$\int\frac{1}{p + q \tan x} \text{ dx }$
Ex. 19.24 | Q 5 | Page 122
$\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }$
Ex. 19.24 | Q 6 | Page 122
$\int\frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx$
Ex. 19.24 | Q 7 | Page 122
$\int\frac{1}{3 + 4 \cot x} dx$
Ex. 19.24 | Q 8 | Page 122
$\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }$
Ex. 19.24 | Q 9 | Page 122
$\int\frac{1}{4 + 3 \tan x} dx$
Ex. 19.24 | Q 10 | Page 122
$\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }$
Ex. 19.24 | Q 11 | Page 122
$\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }$

#### Chapter 19: Indefinite Integrals Exercise 19.25 solutions [Pages 133 - 134]

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

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

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

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

Evaluate the following integrals:

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

Ex. 19.25 | Q 28 | Page 133

Evaluate the following integrals:

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

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

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

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

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

#### Chapter 19: Indefinite Integrals Exercise 19.26 solutions [Pages 14 - 143]

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

Evaluate the following integrals:

$\int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx$

#### Chapter 19: Indefinite Integrals Exercise 19.27 solutions [Page 149]

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

Evaluate the following integrals:

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

#### Chapter 19: Indefinite Integrals solutions [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
$\int\sqrt{2x - x^2} \text{ dx}$

#### Chapter 19: Indefinite Integrals solutions [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
$\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx$

#### Chapter 19: Indefinite Integrals solutions [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
$\int\frac{( x^2 + 1) ( x^2 + 4)}{( x^2 + 3) ( x^2 - 5)} dx$

#### Chapter 19: Indefinite Integrals solutions [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:

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

#### Chapter 19: Indefinite Integrals solutions [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
$\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 9}} \text{ dx}$

#### Chapter 19: Indefinite Integrals Exercise Very Short Answers solutions [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 : $\int\frac{1}{x(1 + \log x)} \text{ dx}$

#### Chapter 19: Indefinite Integrals Exercise MCQ solutions [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

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

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

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

• $a = \frac{1}{10}, b = \frac{2}{5}$

#### Chapter 19: Indefinite Integrals Exercise Revision Excercise solutions [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
$\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}$

#### Chapter 19: Indefinite Integrals Exercise Revision Excercise solutions [Page 203]

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

## Chapter 19: Indefinite Integrals

Ex. 19.1Ex. 19.2Ex. 19.3Ex. 19.4Ex. 19.5Ex. 19.6Ex. 19.7Ex. 19.8Ex. 19.85Ex. 19.9Ex. 19.10Ex. 19.1119..11Ex. 19.12Ex. 19.13Ex. 19.14Ex. 19.15Ex. 19.16Ex. 19.17Ex. 19.18Ex. 19.19Ex. 19.20Ex. 19.21Ex. 19.22Ex. 19.23Ex. 19.24Ex. 19.25Ex. 19.26Ex. 19.27OthersVery Short AnswersMCQRevision Excercise

## RD Sharma solutions for Class 12 Mathematics 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 Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics 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.

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