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R.D. Sharma solutions for Class 12 Mathematics chapter 19 - Indefinite Integrals

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

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R.D. Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Chapter 19 - Indefinite Integrals

Page 4

Q 1.1 | Page 4

Evaluate of the following integral:

(i)  \[\int x^4 dx\]

 

Q 1.2 | Page 4

Evaluate of the following integral: 

\[\int x^\frac{5}{4} dx\]
Q 1.3 | Page 4

Evaluate of the following integral: 

\[\int\frac{1}{x^5}dx\]
Q 1.4 | Page 4

Evaluate of the following integral: 

\[\int\frac{1}{x^{3/2}}dx\]
Q 1.5 | Page 4

Evaluate of the following integral: 

\[\int 3^x dx\]
Q 1.6 | Page 4

Evaluate of the following integral:

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

Evaluate of the following integral:

\[\int 3^{2 \log_3} {}^x dx\]
Q 1.8 | Page 4

Evaluate of the following integral:

\[\int \log_x xdx\] 
Q 2.1 | Page 4

Evaluate: 

\[\int\sqrt{\frac{1 + \cos 2x}{2}}dx\]
Q 2.2 | Page 4

Evaluate:

\[\int\sqrt{\frac{1 - \cos 2x}{2}}dx\]
Q 3 | Page 4

Evaluate : 

\[\int\frac{e^{6 \log_e x} - e^{5 \log_e x}}{e^{4 \log_e x} - e^{3 \log_e x}}dx\]
Q 4 | Page 4

Evaluate: 

\[\int\frac{1}{a^x b^x}dx\]
Q 5.1 | Page 4

Evaluate:

\[\int\frac{\cos 2x + 2 \sin^2 x}{\sin^2 x}dx\]
Q 5.2 | Page 4

Evaluate: 

\[\int\frac{2 \cos^2 x - \cos 2x}{\cos^2 x}dx\]
Q 6 | Page 4

Evaluate:

\[\int\frac{e\log \sqrt{x}}{x}dx\]

Pages 14 - 15

Q 1 | Page 15
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
Q 1 | Page 14
\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]
Q 2 | Page 14
\[\int\left( 2^x + \frac{5}{x} - \frac{1}{x^{1/3}} \right)dx\]
Q 3 | Page 14
\[\int\left\{ \sqrt{x}\left( a x^2 + bx + c \right) \right\} dx\]
Q 4 | Page 14
\[\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx\]
Q 5 | Page 14
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
Q 6 | Page 14
\[\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx\]
Q 7 | Page 14
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\] 
Q 8 | Page 14

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

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

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

 

Q 16 | Page 15
\[\int\frac{\left( x + 1 \right)\left( x - 2 \right)}{\sqrt{x}} dx\]
Q 17 | Page 15
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
Q 18 | Page 15
\[\int \left( 3x + 4 \right)^2 dx\]
Q 20 | Page 15
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
Q 21 | Page 15
\[\int\frac{\sin^2 x}{1 + \cos x} dx\]
Q 22 | Page 15
\[\int\left( \sec^2 x + {cosec}^2 x \right) dx\]
Q 23 | Page 15
\[\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx\]
Q 24 | Page 15
\[\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx\]
Q 25 | Page 15
\[\int \left( \tan x + \cot x \right)^2 dx\]
Q 26 | Page 15
\[\int\frac{1 - \cos 2x}{1 + \cos 2x} dx\]
Q 27 | Page 15
\[\int\frac{\cos x}{1 - \cos x} dx or \int\frac{\cot x}{cosec x - \cot x} dx\]
Q 28 | Page 15
\[\int\frac{\cos^2 x - \sin^2 x}{\sqrt{1} + \cos 4x} dx\]
Q 29 | Page 15
\[\int\frac{1}{1 - \cos x} dx\]
Q 30 | Page 15
\[\int\frac{1}{1 - \sin x} dx\]
Q 31 | Page 15
\[\int\frac{\tan x}{\sec x + \tan x} dx\]
Q 32 | Page 15
\[\int\frac{cosec x}{cosec x - \cot x} dx\]
Q 33 | Page 15
\[\int\frac{1}{1 + \cos 2x} dx\]
Q 34 | Page 15
\[\int\frac{1}{1 - \cos 2x} dx\]
Q 35 | Page 15
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
Q 36 | Page 15
\[\int \cos^{- 1} \left( \sin x \right) dx\]
Q 37 | Page 15
\[\int \cot^{- 1} \left( \frac{\sin 2x}{1 - \cos 2x} \right) dx\]
Q 38 | Page 15
\[\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx\]
Q 39 | Page 15
\[\int\frac{\left( x^3 + 8 \right)\left( x - 1 \right)}{x^2 - 2x + 4} dx\]
Q 40 | Page 15
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
Q 41 | Page 15
\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]
Q 42 | Page 15
\[\int\frac{\cos x}{1 + \cos x} dx\]
Q 43 | Page 15
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
Q 45 | Page 15

If f' (x) = x − \[\frac{1}{x^2}\] \[\frac{1}{2},\]

 

Q 46 | Page 15

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

Q 47 | Page 15

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

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

 

Pages 23 - 24

Q 1 | Page 23
\[\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} dx\]
Q 2 | Page 23
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
Q 3 | Page 23
\[\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx\]
Q 4 | Page 23
\[\int\frac{x + 3}{\left( x + 1 \right)^4} dx\]
Q 5 | Page 23
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
Q 6 | Page 23
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
Q 7 | Page 23
\[\int\frac{2x}{\left( 2x + 1 \right)^2} dx\]
Q 8 | Page 23
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
Q 9 | Page 23
\[\int\sin x\sqrt{1 + \cos 2x} dx\]
Q 10 | Page 23
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
Q 11 | Page 23
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
Q 12 | Page 23
\[\int\frac{1}{1 - \sin\frac{x}{2}} dx\]
Q 13 | Page 23
\[\int\frac{1}{1 + \cos 3x} dx\]
Q 15 | Page 23
\[\int \left( e^x + 1 \right)^2 e^x dx\]
Q 16 | Page 23
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
Q 17 | Page 23
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
Q 18 | Page 23

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

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

Page 30

Q 1 | Page 30

\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]

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

Pages 33 - 196

Q 1 | Page 33
\[\int\frac{x + 1}{\sqrt{2x + 3}} dx\]
Q 2 | Page 33
\[\ intx\sqrt{x + 2} dx\]
Q 3 | Page 33
\[\int\frac{x - 1}{\sqrt{x + 4}} dx\]
Q 4 | Page 33
\[\int\left( x + 2 \right) \sqrt{3x + 5} dx\]
Q 5 | Page 33
\[\int\frac{2x + 1}{\sqrt{3x + 2}} dx\]
Q 5 | Page 196
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} dx\]
Q 6 | Page 33
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
Q 7 | Page 33
\[\int\frac{x}{\sqrt{x + 4}} dx\]
Q 8 | Page 33
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
Q 9 | Page 33
\[\int\left( 5x + 3 \right) \sqrt{2x - 1} dx\]
Q 10 | Page 33
\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]

Pages 3 - 36

Q 1 | Page 36
\[\int \sin^2 \left( 2x + 5 \right) dx\]
Q 3 | Page 36
\[\int \sin^3 \left( 2x + 1 \right) dx\]
Q 4 | Page 36
\[\int \sin^2 b x dx\]
Q 5 | Page 36
\[\int \sin^2 \frac{x}{2} dx\]
Q 6 | Page 3
\[\int \cos^2 \frac{x}{2} dx\]

 

Q 7 | Page 36
\[\int \cos^2 nx dx\]
Q 8 | Page 36
\[\int\sin x \sqrt{1 - \cos 2x} dx\]

Page 38

Q 1 | Page 38
\[\int\sin 4x \cos 7x dx\]
Q 2 | Page 38
\[\int\cos 3x \cos 4x dx\]
Q 3 | Page 38
\[\int\cos mx \cos nx dx m \neq n\]

 

Q 4 | Page 38
\[\int\sin mx \cos nx dx m \neq n\]
Q 5 | Page 38

Integrate the following integrals:

\[\int\sin2x \sin4x \sin6x dx\]
Q 6 | Page 38

Integrate the following integrals:

\[\int\ sin\ x \cos2x \sin3x\ dx\]

Pages 47 - 48

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

Evaluate the following integrals: 

\[\int\frac{\sec x}{\sec 2x}dx\]
Q 6 | Page 47
\[\int\frac{\cos 2x}{\left( \cos x + \sin x \right)^2} dx\]
Q 7 | Page 47
\[\int\frac{\sin \left( x - a \right)}{\sin \left( x - b \right)} dx\]
Q 8 | Page 47
\[\int\frac{\sin \left( x - \alpha \right)}{\sin \left( x + \alpha \right)} dx\]
Q 9 | Page 47
\[\int\frac{1 + \tan x}{1 - \tan x} dx\]
Q 10 | Page 47
\[\int\frac{\cos x}{\cos \left( x - a \right)} dx\] 
Q 11 | Page 47
\[\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx\]
Q 12 | Page 47
\[\int\frac{e^{3x}}{e^{3x} + 1} dx\]
Q 13 | Page 47
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
Q 14 | Page 47
\[\int\frac{1 - \cot x}{1 + \cot x} dx\]
Q 15 | Page 47
\[\int\frac{\sec x cosec x}{\log (\tan x)} dx\]
Q 16 | Page 47
\[\int\frac{1}{x (3 + \log x)} dx\]
Q 17 | Page 47
\[\int\frac{e^x + 1}{e^x + x} dx\]
Q 18 | Page 47
\[\int\frac{1}{x \log x} dx\]
Q 19 | Page 47
\[\int\frac{\sin 2x}{a \cos^2 x + b \sin^2 x} dx\]
Q 20 | Page 47
\[\int\frac{\cos x}{2 + 3 \sin x} dx\]
Q 21 | Page 47
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
Q 22 | Page 47
\[\int\frac{a}{b + c e^x} dx\]
Q 23 | Page 47
\[\int\frac{1}{e^x + 1} dx\]
Q 24 | Page 47
\[\int\frac{\cot x}{\log \sin x} dx\]
Q 25 | Page 47
\[\int\frac{e^{2x}}{e^{2x} - 2} dx\]
Q 26 | Page 47
\[\int\frac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x} dx\]
Q 27 | Page 48
\[\int\frac{\cos 2x + x + 1}{x^2 + \sin 2x + 2x} dx\]
Q 28 | Page 48
\[\int\frac{1}{\cos\left( x + a \right) \cos\left( x + b \right)}dx\]
Q 29 | Page 48
\[\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx\]
Q 30 | Page 48
\[\int\frac{\cos 4x - \cos 2x}{\sin 4x - \sin 2x} dx\]
Q 31 | Page 48
\[\int\frac{sec x}{\log \left( \sec x + \tan x \right)} dx\]
Q 32 | Page 48
\[\int\frac{cosec x}{\log \tan\frac{x}{2}} dx\]
Q 33 | Page 48
\[\int\frac{1}{x \log x \log \left( \log x \right)} dx\]
Q 34 | Page 48
\[\int\frac{{cosec}^2 x}{1 + \cot x} dx\]
Q 35 | Page 48
\[\int\frac{10 x^9 + {10}^x \log_e 10}{{10}^x + x^{10}} dx\]
Q 36 | Page 48
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
Q 37 | Page 48
\[\int\frac{1 + \tan x}{x + \log \sec x} dx\]
Q 38 | Page 48
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x} dx\]
Q 39 | Page 48
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
Q 40 | Page 48
\[\int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)} dx\]
Q 41 | Page 48
\[\int\frac{\sec^2 x}{\tan x + 2} dx\]
Q 42 | Page 48
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
Q 43 | Page 48
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
Q 44 | Page 48
\[\int\frac{1 + \cot x}{x + \log \sin x} dx\]
Q 45 | Page 48
\[\int\frac{1}{\sqrt{x}\left( \sqrt{x} + 1 \right)} dx\]
Q 46 | Page 48
\[\int\tan 2x \tan 3x \tan 5x dx\]
Q 47 | Page 48
\[\int\left\{ 1 + \tan x \tan \left( x + \theta \right) \right\} dx\]
Q 48 | Page 48
\[\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx\]
Q 49 | Page 48
\[\int\frac{e^{x - 1} + x^{e - 1}}{e^x + x^e} dx\]
Q 50 | Page 48
\[\int\frac{1}{\sin x \cos^2 x} dx\]
Q 51 | Page 48
\[\int\frac{1}{\cos 3x - \cos x} dx\]

Pages 57 - 149

Q 1 | Page 57
\[\int\frac{\log x}{x} dx\]
Q 2 | Page 57
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
Q 3 | Page 57
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
Q 4 | Page 57
\[\int\sqrt{1 + e^x} e^x dx\]
Q 5 | Page 57
\[\int\sqrt[3]{\cos^2 x}\sin x dx\]
Q 6 | Page 57
\[\int\frac{e^x}{\left( 1 + e^x \right)^2} dx\]
Q 7 | Page 57
\[\int \cot^3 x {cosec}^2 x dx\]
Q 8 | Page 57

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

Q 9 | Page 57
\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}} dx\]
Q 9 | Page 149
\[\int\frac{1}{x^3}\sin \left( \log x \right) dx\]
Q 10 | Page 57
\[\int\frac{1}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2} dx\]
Q 11 | Page 58
\[\int\frac{\cot x}{\sqrt{\sin x}} dx\]
Q 12 | Page 58
\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]
Q 13 | Page 58
\[\int\frac{\cos^3 x}{\sqrt{\sin x}} dx\]
Q 14 | Page 58
\[\int\frac{\sin^3 x}{\sqrt{\cos x}} dx\]
Q 15 | Page 58
\[\int\frac{1}{\sqrt{\tan^{- 1} x} . \left( 1 + x^2 \right)} dx\]
Q 17 | Page 58
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
Q 18 | Page 58
\[\int \sin^5 x \cos x dx\]
Q 19 | Page 58
\[\int \tan^{3/2} x \sec^2 x dx\]
Q 20 | Page 58
\[\int\frac{x^3}{\left( x^2 + 1 \right)^3} dx\]
Q 21 | Page 58
\[\int\left( 4x + 2 \right)\sqrt{x^2 + x + 1} dx\]
Q 22 | Page 58
\[\int\frac{4x + 3}{\sqrt{2 x^2 + 3x + 1}} dx\]
Q 23 | Page 58
\[\int\frac{1}{1 + \sqrt{x}} dx\]
Q 24 | Page 58
\[\int e^\ cos^2 x \sin2x dx\]
Q 25 | Page 58
\[\int\frac{1 + \cos x}{\left( x + \sin x \right)^3} dx\]
Q 26 | Page 58
\[\int\frac{\cos x - \sin x}{1 + \sin 2x} dx\]
Q 27 | Page 58
\[\int\frac{\sin 2x}{\left( a + b \cos 2x \right)^2} dx\]
Q 28 | Page 58
\[\int\frac{\log x^2}{x} dx\]
Q 29 | Page 58
\[\int\frac{\sin x}{\left( 1 + \cos x \right)^2} dx\]

 

Q 30 | Page 58
\[\int\cot x \cdot \log \sin x dx\]
Q 31 | Page 58
\[\int\sec x \cdot \log \left( \sec x + \tan x \right) dx\]
Q 32 | Page 58
\[\ intcosec x \log \left( cosec x - \cot x \right) dx\]
Q 33 | Page 58
\[\int x^3 \cos x^4 dx\]
Q 34 | Page 58
\[\int x^3 \sin x^4 dx\]
Q 35 | Page 58
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
Q 36 | Page 58
\[\int x^3 \sin \left( x^4 + 1 \right) dx\]
Q 37 | Page 58
\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x e^x \right)} dx\]
Q 38 | Page 58
\[\int x^2 e^{x^3} \cos \left( e^{x^3} \right) dx\]
Q 39 | Page 58
\[\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx\]
Q 40 | Page 58
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
Q 41 | Page 58
\[\int\tan x \sec^2 x\sqrt{1 - \tan^2 x} dx\]
Q 42 | Page 58
\[\int\log x\frac{\sin \left\{ 1 + \left( \log x \right)^2 \right\}}{x} dx\]
Q 43 | Page 58
\[\int\frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) dx\]
Q 44 | Page 58
\[\int \sec^4 x \tan x dx\]
Q 45 | Page 58
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
Q 46 | Page 58
\[\int\frac{\cos^5 x}{\sin x} dx\]
Q 47 | Page 59
\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
Q 48 | Page 59
\[\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( x e^x \right)} dx\]
Q 49 | Page 59
\[\int 5^{x + \tan^{- 1} x} . \left( \frac{x^2 + 2}{x^2 + 1} \right) dx\]
Q 50 | Page 59
\[\int\frac{e^{m \sin^{- 1} x}}{\sqrt{1 - x^2}} dx\]
Q 51 | Page 59
\[\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
Q 52 | Page 59
\[\int\frac{\sin \left( \tan^{- 1} x \right)}{1 + x^2} dx\]
Q 53 | Page 59
\[\int\frac{\sin \left( \log x \right)}{x} dx\]
Q 54 | Page 59
\[\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx\]
Q 55 | Page 59
\[\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx\]
Q 56 | Page 59
\[\ intx\frac{\tan^{- 1} x^2}{1 + x^4} dx\]
Q 57 | Page 59
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx\]

 

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

Page 65

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

 

Q 9 | Page 65
\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
Q 10 | Page 65
\[\int\frac{1}{x^\ sfrac{1}{3} \left( x^\ sfrac{1}{3} - 1 \right)}dx\]

Page 69

Q 1 | Page 69
\[\int \tan^3 x \sec^2 x dx\]
Q 2 | Page 69
\[\int\tan x \sec^4 x dx\]
Q 3 | Page 69
\[\int \tan^5 x \sec^4 x dx\]
Q 4 | Page 69
\[\int \sec^6 x \tan x dx\]
Q 5 | Page 69
\[\int \tan^5 x dx\]
Q 6 | Page 69
\[\int\sqrt{\tan x} \sec^4 x dx\]
Q 7 | Page 69
\[\int \sec^4 2x dx\]
Q 8 | Page 69
\[\int {cosec}^4 3x dx\]
Q 9 | Page 69
\[\int \cot^n {cosec}^2 x dx, n \neq - 1\]
Q 10 | Page 69
\[\int \cot^5 x {cosec}^4 x dx\]
Q 11 | Page 69
\[\int \cot^5 x dx\]
Q 12 | Page 69
\[\int \cot^6 x dx\]

Page 73

Q 1 | Page 73
\[\int \sin^4 x \cos^3 x dx\]
Q 2 | Page 73
\[\int \sin^5 x dx\]
Q 3 | Page 73
\[\int \cos^5 x dx\]
Q 4 | Page 73
\[\int \sin^5 x \cos x dx\]
Q 5 | Page 73
\[\int \sin^3 x \cos^6 x dx\]
Q 6 | Page 73
\[\int \cos^7 x dx\]
Q 7 | Page 73
\[\int x \cos^3 x^2 \sin x^2 dx\]
Q 8 | Page 73
\[\int \sin^7 x dx\]
Q 9 | Page 73
\[\int \sin^3 x \cos^5 x dx\]
Q 10 | Page 73
\[\int\frac{1}{\sin^4 x \cos^2 x} dx\]
Q 11 | Page 73
\[\int\frac{1}{\sin^3 x \cos^5 x} dx\]
Q 12 | Page 73
\[\int\frac{1}{\sin^3 x \cos x} dx\]
Q 13 | Page 73
\[\int\frac{1}{\sin x \cos^3 x} dx\]

Page 79

Q 1 | Page 79
Evaluate the following integrals:
\[\int\frac{x^2}{\left( a^2 - x^2 \right)^\ sfrac\ {3}{2}}dx\]
Q 2 | Page 79

Evaluate the following integrals:

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

Evaluate the following integrals:

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

Evaluate the following integrals:

\[\int\frac{\sqrt{1 + x^2}}{x^4}dx\]
Q 5 | Page 79

Evaluate the following integrals:

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

 

Page 83

Q 1 | Page 83
\[\int\frac{1}{a^2 - b^2 x^2} dx\]
Q 2 | Page 83
\[\int\frac{1}{a^2 x^2 - b^2} dx\]
Q 3 | Page 83
\[\int\frac{1}{a^2 x^2 + b^2} dx\]
Q 4 | Page 83
\[\int\frac{x^2 - 1}{x^2 + 4} dx\]
Q 5 | Page 83
\[\int\frac{1}{\sqrt{1 + 4 x^2}} dx\]

 

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

Page 86

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

Pages 16 - 90

Q 1 | Page 90
\[\int\frac{\sec^2 x}{1 - \tan^2 x} dx\]
Q 2 | Page 90
\[\int\frac{e^x}{1 + e^{2x}} dx\]
Q 3 | Page 90
\[\int\frac{\cos x}{\sin^2 x + 4 \sin x + 5} dx\]
Q 4 | Page 90
\[\int\frac{e^x}{e^{2x} + 5 e^x + 6} dx\]
Q 5 | Page 90
\[\int\frac{e^{3x}}{4 e^{6x} - 9} dx\]
Q 6 | Page 90
\[\int\frac{dx}{e^x + e^{- x}}\]
Q 7 | Page 90
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
Q 8 | Page 90
\[\int\frac{3 x^5}{1 + x^{12}} dx\]
Q 9 | Page 90
\[\int\frac{x^2 dx}{x^6 - a^6} dx\]
Q 10 | Page 90
\[\int\frac{x^2}{x^6 + a^6} dx\]
Q 11 | Page 90
\[\int\frac{1}{x \left( x^6 + 1 \right)} dx\]
Q 12 | Page 90
\[\int\frac{x}{x^4 - x^2 + 1} dx\]
Q 13 | Page 90
\[\int\frac{x}{3 x^4 - 18 x^2 + 11} dx\]
Q 14 | Page 90
\[\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx\]
Q 15 | Page 90
\[\int\frac{1}{\ cosx + cosecx}dx\]
Q 24 | Page 16
\[\int\limits_0^{\pi/2} \sqrt{1 + \sin x}\ dx\]

Page 93

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

Pages 98 - 99

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

Page 104

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

Evaluate the following integrals:

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

Page 106

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

Pages 110 - 111

Q 1 | Page 110
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} dx\]
Q 2 | Page 110
\[\int\frac{2x + 1}{\sqrt{x^2 + 2x - 1}} dx\]
Q 3 | Page 110
\[\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} dx\]
Q 4 | Page 110
\[\int\frac{6x - 5}{\sqrt{3 x^2 - 5x + 1}} dx\]
Q 6 | Page 110
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} dx\]
Q 7 | Page 110
\[\int\frac{x + 2}{\sqrt{x^2 + 2x - 1}} dx\]
Q 8 | Page 110
\[\int\frac{x + 2}{\sqrt{x^2 - 1}} dx\]
Q 9 | Page 110
\[\int\frac{x - 1}{\sqrt{x^2 + 1}} dx\]
Q 10 | Page 110
\[\int\frac{x}{\sqrt{x^2 + x + 1}} dx\]
Q 11 | Page 110
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
Q 12 | Page 110
\[\int\frac{2x + 5}{\sqrt{x^2 + 2x + 5}} dx\]
Q 13 | Page 110
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} dx\]
Q 14 | Page 110
\[\int\sqrt{\frac{1 - x}{1 + x}} dx\]
Q 15 | Page 111
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} dx\]
Q 16 | Page 111
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} dx\]
Q 17 | Page 111
\[\int\frac{5x + 3}{\sqrt{x^2 + 4x + 10}} dx\]
Q 18 | Page 111

Evaluate the following integrals: 

\[\int\frac{x + 2}{\sqrt{x^2 + 2x + 3}}dx\]

Page 114

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

Page 117

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

Page 122

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

Pages 133 - 143

Q 1 | Page 133
\[\int x \cos x\ dx\]
Q 2 | Page 133
\[\int\log \left( x + 1 \right) dx\]
Q 3 | Page 133
\[\int x^3 \log x dx\]
Q 4 | Page 133
\[\int x e^x dx\]
Q 5 | Page 143
\[\int e^x \left( \frac{x - 1}{2 x^2} \right) dx\]
Q 5 | Page 133
\[\int x e^{2x} dx\]
Q 6 | Page 133
\[\int x^2 e^{- x} dx\]
Q 7 | Page 133
\[\int x^2 \cos x dx\]
Q 8 | Page 133
\[\int x^2 \cos 2x\ dx\]
Q 9 | Page 133
\[\int x \sin 2x dx\]
Q 10 | Page 133
\[\int\frac{\log \left( \log x \right)}{x} dx\]
Q 11 | Page 133
\[\int x^2 \cos x dx\]
Q 12 | Page 133
\[\int x\ {cosec}^2 x\ dx\]
Q 13 | Page 133
\[\int x \cos^2 x\ dx\]
Q 14 | Page 133
\[\int x^n \cdot \log x dx\]
Q 15 | Page 133
\[\int\frac{\log x}{x^n} dx\]
Q 16 | Page 133
\[\int x^2 \sin^2 x\ dx\]
Q 17 | Page 133
\[\int2 x^3 e^{x^2} dx\]
Q 18 | Page 133
\[\int x^3 \cos x^2 dx\]
Q 19 | Page 133
\[\int x \sin x \cos x\ dx\]

 

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

Evaluate the following integrals:

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

 

Q 28 | Page 133

Evaluate the following integrals:

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

 

Q 29 | Page 133
\[\int {cosec}^3 x\ dx\]
Q 30 | Page 133
\[\int \sec^{- 1} \sqrt{x}\ dx\]
Q 31 | Page 134
\[\int \sin^{- 1} \sqrt{x} dx\]
Q 32 | Page 134
\[\int x \tan^2 x\ dx\]
= ‚Äč∫ x (sec2 x – 1) dx
Q 33 | Page 134
\[\int x\left( \frac{\sec 2x - 1}{\sec 2x + 1} \right) dx\]
Q 34 | Page 134
\[\int\left( x + 1 \right) e^x \log \left( x e^x \right) dx\]
Q 35 | Page 134
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) dx\]
Q 36 | Page 134
\[\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) dx\]
Q 37 | Page 134
\[\int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]
Q 38 | Page 134
\[\int x^2 \sin^{- 1} x\ dx\]
Q 39 | Page 134
\[\int\frac{\sin^{- 1} x}{x^2} dx\]
Q 40 | Page 134
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} dx\]
Q 41 | Page 134
\[\int \cos^{- 1} \left( 4 x^3 - 3x \right) dx\]
Q 42 | Page 134
\[\int \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx\]
Q 43 | Page 134
\[\int \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
Q 44 | Page 134
\[\int\left( x + 1 \right) \log x dx\]
Q 45 | Page 134
\[\int x^2 \tan^{- 1} x\ dx\]
Q 46 | Page 134

\[\int\left( e^ (\log x) + \sin x \right) \cos x dx\]

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

Pages 14 - 143

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

Evaluate the following integrals:

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

Page 149

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

Evaluate the following integrals:

\[\int e^{2x} \sin\left( 3x + 1 \right) dx\]
Q 8 | Page 149
\[\int e^x \sin^2 x\ dx\]
Q 10 | Page 149
\[\int e^{2x} \cos^2 x\ dx\]
Q 11 | Page 149
\[\int e^{- 2x} \sin x\ dx\]
Q 12 | Page 149
\[\int x^2 e^{x^3} \cos x^3 dx\]

Pages 154 - 155

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

Pages 158 - 159

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

 

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

Evaluate the following integrals:

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

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
\[\int\frac{a x^2 + bx + c}{\left( x - a \right) \left( x - b \right) \left( x - c \right)} dx, where\ a, b, c\ are\ distinct\]
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
\[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
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

Q 1 | Page 190
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} dx\]
Q 2 | Page 190
\[\int\sqrt{\cot \theta}d \theta\]
Q 3 | Page 190
\[\int\frac{x^2 + 9}{x^4 + 81} dx\]

 

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

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

Pages 179 - 198

Q 1 | Page 197

Write a value of

\[\int\frac{1 + \cot x}{x + \log \sin x} dx\]
Q 2 | Page 197

Write a value of

\[\int e^{3 \log x} x^4\ dx\]
Q 3 | Page 197

Write a value of

\[\int x^2 \sin x^3 dx\]
Q 4 | Page 197

Write a value of

\[\int \tan^3 x \sec^2 x dx\]

 

Q 5 | Page 197

Write a value of

\[\int e^x \left( \sin x + \cos x \right) dx\]

 

Q 6 | Page 197

Write a value of

\[\int \tan^6 x \sec^2 x dx\]
Q 7 | Page 197

Write a value of

\[\int\frac{\cos x}{3 + 2 \sin x} dx\]
Q 8 | Page 197

Write a value of

\[\int e^x \sec x \left( 1 + \tan x \right) dx\]
Q 9 | Page 197

Write a value of

\[\int\frac{\log x^n}{x} dx\]
Q 10 | Page 197

Write a value of

\[\int\frac{\left( \log x \right)^n}{x} dx\]
Q 11 | Page 197

Write a value of

\[\int e^{\log\ sin x} \cos x\ dx\]
Q 12 | Page 197
\[\int \sin^3 x \cos x\ dx\]

 

Q 13 | Page 197

Write a value of

\[\int\tan x \sec^3 x\ dx\]
Q 14 | Page 197

Write a value of

\[\int \cos^4 x \sin x dx\]
Q 15 | Page 179

Write a value of

\[\int\frac{1}{1 + e^x} dx\]
Q 16 | Page 197

Write a value of

\[\int\frac{1}{1 + 2 e^x} dx\]
Q 17 | Page 197

Write a value of

\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Q 18 | Page 197

Write a value of

\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Q 19 | Page 197

Write a value of

\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Q 20 | Page 197

Write a value of

\[\int \log_e x\ dx\]

 

Q 21 | Page 197

Write a value of

\[\int a^x e^x dx\]
Q 22 | Page 197

Write a value of

\[\int e^{2 x^2 + \ln x} dx\]
Q 23 | Page 197

Write a value of

\[\int\left( e^{x \log_e a} + e^{a \log_e x} \right) dx\]
Q 24 | Page 197
Write a value of
\[\int\frac{\cos x}{\sin x \log \sin x} dx\]

 

Q 25 | Page 197

Write a value of

\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} dx\]
Q 26 | Page 197

Write a value of

\[\int\frac{a^x}{3 + a^x} dx\]
Q 27 | Page 197

Write a value of

\[\int\frac{1 + \log x}{3 + x \log x} dx\]
Q 28 | Page 197

Write a value of

\[\int\frac{\sin x}{\cos^3 x} dx\]
Q 29 | Page 197

Write a value of

\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} dx\]
Q 30 | Page 197

Write a value of

\[\int\frac{1}{x \left( \log x \right)^n} dx\]
Q 31 | Page 197

Write a value of

\[\int e^{ax} \sin\ bx\ dx\]
Q 32 | Page 197
Write a value of
\[\int e^{ax} \cos\ bx\ dx\]

 

Q 33 | Page 198

Write a value of

\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\]
Q 34 | Page 198

Write a value of

\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\]
Q 35 | Page 198

Write a value of

\[\int\sqrt{4 - x^2} dx\]
Q 36 | Page 198

Write a value of

\[\int\sqrt{9 + x^2} dx\]
Q 37 | Page 198

Write a value of

\[\int\sqrt{x^2 - 9} dx\]
Q 38 | Page 198

Evaluate:

\[\int\frac{x^2}{1 + x^3} dx\]
Q 39 | Page 198

Evaluate:

\[\int\frac{x^2 + 4x}{x^3 + 6 x^2 + 5} dx\]
Q 40 | Page 198

Evaluate:

\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx\]

 

Q 41 | Page 198

Evaluate:

\[\int\frac{\sin \sqrt{x}}{\sqrt{x}} dx\]
Q 42 | Page 198

Evaluate:

\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\]
Q 43 | Page 198

Evaluate:

\[\int\frac{\left( 1 + \log x \right)^2}{x} dx\]
Q 44 | Page 198

Evaluate:

\[\int \sec^2 \left( 7 - 4x \right) dx\]
Q 45 | Page 198

Evaluate:

\[\int\frac{\log x}{x} dx\]
Q 46 | Page 198

Evaluate:

\[\int 2^x dx\]
Q 47 | Page 198

Write a value of

\[\int\frac{1 - \sin x}{\cos^2 x} dx\]
Q 48 | Page 198

Evaluate:

\[\int\frac{x^3 - 1}{x^2} dx\]
Q 49 | Page 198

Evaluate:

\[\int\frac{x^3 - x^2 + x - 1}{x - 1} dx\]
Q 50 | Page 198

Evaluate:

\[\int\frac{e\tan^{- 1} x}{1 + x^2} dx\]
Q 51 | Page 198

Evaluate:

\[\int\frac{1}{\sqrt{1 - x^2}} dx\]
Q 52 | Page 198

Write the value of

\[\int\sec x \left( \sec x + \tan x \right) dx\]
Q 53 | Page 198

Evaluate: 

\[\int\frac{1}{x^2 + 16}dx\]
Q 54 | Page 198

Evaluate: 

\[\int\left( 1 - x \right)\sqrt{x}\ dx\]
Q 55 | Page 198

Evaluate: 

\[\int\frac{x + \cos6x}{3 x^2 + \sin6x}dx\]
Q 56 | Page 198
\[If \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, then\ write\ the\ value\ of\ f\left( x \right) .\]
Q 57 | Page 198
\[If \int e^x \left( \tan x + 1 \right)\sec x dx = e^x f\left( x \right) + C, then\ write\ the\ value\ of\ \ f\left( x \right) .\]

 

 

Q 58 | Page 198

Evaluate: 

\[\int\frac{2}{1 - \cos2x}dx\]
Q 59 | Page 198

Write the anti-derivative of 

\[\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .\]
Q 60 | Page 198

Evaluate: 

\[\int \cos^{- 1} \left( \sin x \right) dx\]
Q 61 | Page 198

Evaluate: 

\[\int\frac{1}{\sin^2 x \cos^2 x}dx\]
Q 62 | Page 198

Evaluate : 

\[\int\frac{1}{x(1 + \log x)} dx\]

Pages 198 - 203

Q 1 | Page 198
\[\int\frac{x}{4 + x^4} dx\] is equal to
(a) \[\frac{1}{4} \tan^{- 1} x^2 + C\]
(b) \[\frac{1}{4} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
(c) \[\frac{1}{2} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
(d) none of these

 

Q 2 | Page 198
\[\int\frac{1}{\cos x + \sqrt{3} \sin x} dx\] is equal to
(a) \[\int\frac{1}{\cos x + \sqrt{3} \sin x} dx\]
(b) \[\log \tan \left( \frac{x}{2} - \frac{\pi}{3} \right) + C\]
(c)\[\log \tan \left( \frac{x}{2} - \frac{\pi}{3} \right) + C\]
(d) none of these
Q 3 | Page 200
\[\int x \sec x^2 dx\ is\ equal\ to\]

(a) \[\frac{1}{2}\] log (sec x2 + tan x2) + C

(b)\[\frac{x^2}{2}\]  log (sec x2 + tan x2) + C

(c) 2 log (sec x2 + tan x2) + C

(d) none of these

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) A =\[\frac{2}{3}\], B =\[\frac{5}{3}\]

(b) A =\[\frac{1}{3}\], B = \[\frac{2}{3}\]

(c) A =\[- \frac{2}{3}\], B =\[\frac{5}{3}\]

(d) A =\[\frac{1}{3}\], B =\[- \frac{5}{3}\]

Q 5 | Page 200
\[\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx\ is\ equal\ to\]

(a) xsin x + C
(b) xsin x cos x + C
(c) \[\frac{\left( x^{\sin x} \right)^2}{2} + C\]
(d) none of these

Q 6 | Page 200

Integration of

\[\frac{1}{1 + \left( \log_e x \right)^2}\] with respect to loge x is

(a)\[\frac{\tan^{- 1} \left( \log_e x \right)}{x} + C\]

(b)\[\tan^{- 1} \left( \log_e x \right) + C\]

(c)\[\frac{\tan^{- 1} x}{x} + C\]

(d) none of these

Q 7 | Page 200

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

(a)\[- \frac{1}{16}\]

(b)\[\frac{1}{8}\]

(c)\[\frac{1}{16}\]

(d)\[- \frac{1}{8}\]

Q 8 | Page 200

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

(a) −1/2 (b) 1/2 (c) −1 (d) 1

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

(b) xex + C

(c) − xex + C

(d) xex + C
Q 10 | Page 200

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

(a) \[- \frac{1}{\log_e 2}\]

(b) − loge 2

(c) − 1

(d)\[\frac{1}{2}\]

Q 11 | Page 200
\[\int\frac{1}{1 + \tan x} dx =\]

(a) loge (x + sin x) + C

(b) loge (sin x + cos x) + C

(c) \[2 \sec^2 \frac{x}{2} + C\]

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

Q 12 | Page 200
\[\int \left| x \right|^3 dx\]
(a) \[\frac{- x^4}{4} + C\]
(b) \[\frac{\left| x \right|^4}{4} + C\]
(c) \[\frac{x^4}{4} + C\]

(d) none of these

Q 13 | Page 200

The value of

\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\]
(a) 2 cos \[\sqrt{x}\]
(b) \[\sqrt{\frac{\cos x}{x}} + C\]
(c) sin \[\sqrt{x} + C\]
(d) 2 sin \[\sqrt{x} + C\] 
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

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

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

\[\frac{1}{\sqrt{3}} \tan^{- 1} \left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + C\]

\[\frac{1}{4} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C\]
\[\frac{1}{7} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C\]
Q 18 | Page 201
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
\[\log\left| 1 + \cot\frac{x}{2} \right| + C\]
\[\log\left| 1 - \tan\frac{x}{2} \right| + C\]
\[\log\left| 1 - \cot\frac{x}{2} \right| + C\]
\[\log\left| 1 + \tan\frac{x}{2} \right| + C\]
Q 19 | Page 201
\[\int\frac{\sin x}{3 + 4 \cos^2 x} dx\]

log (3 + 4 cos2 x) + C

\[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{\cos x}{\sqrt{3}} \right) + C\]
\[- \frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
\[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
Q 20 | Page 201
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[- e^x \tan\frac{x}{2} + C\]
\[- e^x \cot\frac{x}{2} + C\]
\[- \frac{1}{2} e^x \tan\frac{x}{2} + C\]
\[- \frac{1}{2} e^x \cot\frac{x}{2} + C\]
Q 21 | Page 201
\[\int\frac{2}{\left( e^x + e^{- x} \right)^2} dx\]
\[\frac{- e^{- x}}{e^x + e^{- x}} + C\]
\[- \frac{1}{e^x + e^{- x}} + C\]
\[\frac{- 1}{\left( e^x + 1 \right)^2} + C\]
\[\frac{1}{e^x - e^{- x}} + C\]
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

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

Q 24 | Page 202

The primitive of the function

\[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0 is\]

\[\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
\[\log_e a \cdot a^{x + \frac{1}{x}}\]
\[\frac{a^{x + \frac{1}{x}}}{x} \log_e a\]
\[x\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
Q 25 | Page 202

The value of 

\[\int\frac{1}{x + x \log x} dx\] is

1 + log x

log (1 + log x)

x + log x

x log (1 + log x)

Q 26 | Page 202

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

\[\sin^{- 1} \sqrt{x} + C\]
\[\sin^{- 1} \left\{ \sqrt{x} - \sqrt{x \left( 1 - x \right)} \right\} + C\]
\[\sin^{- 1} \left\{ \sqrt{x \left( 1 - x \right)} \right\} + C\]
\[\sin^{- 1} \sqrt{x} - \sqrt{x \left( 1 - x \right)} + C\]
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)

Q 28 | Page 202

The value o

\[\int\frac{\sin x + \cos x}{\sqrt{1 - \sin 2x}} dx\]

 is equal to

\[\sqrt{\sin 2x} + C\]
\[\sqrt{\cos 2x} + C\]

 ± (sin x − cos x) + C

 ± log (sin x − cos x) + C

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

Q 31 | Page 202
\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\] is equal to 
\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\]
\[2\left( \sin x - x\cos\theta \right) + C\]
\[2\left( \sin x - x\cos\theta \right) + C\]
\[2\left( \sin x - 2x\cos\theta \right) + C\]
Q 32 | Page 202
\[\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx\]  is equal to 

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

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

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

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

 

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 

\[ a = \frac{1}{3}, b = 1\]

\[a = - \frac{1}{3}, b = 1\]

\[ a = - \frac{1}{3}, b = - 1\]

\[ a = \frac{1}{3}, b = - 1\]

 

Q 34 | Page 202
\[\int\frac{x^3}{x + 1}dx\] is equal to

 

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

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

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

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

 

Q 35 | Page 203

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

\[ a = - \frac{1}{10}, b = - \frac{2}{5}\]

\[a = \frac{1}{10}, b = - \frac{2}{5}\]

\[ a = - \frac{1}{10}, b = \frac{2}{5}\]

\[ a = \frac{1}{10}, b = \frac{2}{5}\]

Pages 203 - 205

Q 1 | Page 203

\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} dx\]

Q 2 | Page 203

\[\int\frac{1 - x^4}{1 - x} dx\]

Q 3 | Page 203

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

Q 4 | Page 203

\[\int\frac{8x + 13}{\sqrt{4x + 7}} dx\]

Q 5 | Page 203

\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} dx\]

Q 6 | Page 203
\[\int\frac{\left( 2^x + 3^x \right)^2}{6^x} dx\]
Q 7 | Page 203
\[\int\frac{\sin x}{1 + \sin x} dx\]
Q 8 | Page 203
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} dx\]
Q 9 | Page 203
\[\int \sec^2 x \cos^2 2x dx\]
Q 10 | Page 203
\[\int {cosec}^2 x \cos^2 2x dx\]
Q 11 | Page 203
\[\int \sin^4 2x\ dx\]
Q 12 | Page 203
\[\int \cos^3 3x\ dx\]
Q 13 | Page 203
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}\]
Q 14 | Page 203
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} dx\]
Q 15 | Page 203
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx\]
Q 16 | Page 203
\[\int\frac{1}{e^x + 1} dx\]
Q 17 | Page 203
\[\int\frac{e^x - 1}{e^x + 1} dx\]
Q 18 | Page 203
\[\int\frac{1}{e^x + e^{- x}} dx\]
Q 20 | Page 203
\[\int\frac{\cos^7 x}{\sin x} dx\]
Q 21 | Page 203
\[\int\sin x \sin 2x \sin\ 3x\ dx\]
Q 22 | Page 203
\[\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} dx\]
Q 23 | Page 203
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} dx\]
Q 24 | Page 203
\[\int\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} dx\]
Q 25 | Page 203
\[\int\frac{1}{\cos \left( x - a \right) \cos \left( x - b \right)} dx\]
Q 26 | Page 203
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
Q 27 | Page 203
\[\int\frac{\sin x}{\cos 2x} dx\]
Q 28 | Page 203
\[\int \tan^3 x\ dx\]
Q 29 | Page 203
\[\int \tan^4 x\ dx\]
Q 30 | Page 203
\[\int \tan^5 x\ dx\]
Q 31 | Page 203
\[\int \cot^4 x\ dx\]
Q 32 | Page 203
\[\int \cot^5 x\ dx\]
Q 33 | Page 203
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
Q 34 | Page 203
\[\int x\sqrt{2x + 3} dx\]
Q 35 | Page 203
\[\int\frac{x^3}{\left( 1 + x^2 \right)^2} dx\]
Q 36 | Page 203
\[\int x \sin^5 x^2 \cos x^2 dx\]
Q 37 | Page 203
\[\int \sin^3 x \cos^4 x\ dx\]
Q 38 | Page 203
\[\int \sin^5 x\ dx\]
Q 39 | Page 203
\[\int \cos^5 x\ dx\]
Q 40 | Page 203
\[\int\sqrt{\sin x} \cos^3 x\ dx\]
Q 41 | Page 203
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} dx\]
Q 42 | Page 203
\[\int\frac{1}{\sqrt{x^2 - a^2}} dx\]
Q 43 | Page 203
\[\int\frac{1}{\sqrt{x^2 + a^2}} dx\]
Q 44 | Page 203
\[\int\frac{1}{4 x^2 + 4x + 5} dx\]
Q 45 | Page 203
\[\int\frac{1}{x^2 + 4x - 5} dx\]
Q 46 | Page 203
\[\int\frac{1}{1 - x - 4 x^2} dx\]
Q 47 | Page 203
\[\int\frac{1}{3 x^2 + 13x - 10} dx\]
Q 48 | Page 203
\[\int\frac{\sin x}{\sqrt{\cos^2 x - 2 \cos x - 3}} dx\]
Q 49 | Page 204
\[\int\sqrt{cosec x - 1} dx\]
Q 50 | Page 204
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} dx\]
Q 51 | Page 204
\[\int\frac{x + 1}{x^2 + 4x + 5} dx\]
Q 52 | Page 204
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} dx\]
Q 53 | Page 204
\[\int\sqrt{\frac{1 + x}{x}} dx\]
Q 54 | Page 204

\[\int\sqrt{\frac{1 - x}{x}} dx\]

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

\[\int\frac{1}{4 \sin^2 x + 4 \sin x \cos x + 5 \cos^2 x} dx\]

Q 58 | Page 204
\[\int\frac{1}{a + b \tan x} dx\]
Q 59 | Page 204
\[\int\frac{1}{\sin^2 x + \sin 2x} dx\]
Q 60 | Page 204

\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} dx\]

Q 61 | Page 204

\[\int\frac{x^3}{\sqrt{x^8 + 4}} dx\]

Q 62 | Page 204

\[\int\frac{1}{2 - 3 \cos 2x} dx\]

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

\[\int\frac{1}{\sin^4 x + \cos^4 x} dx\]

Q 69 | Page 204
\[\int\frac{1}{5 - 4 \sin x} dx\]
Q 70 | Page 205

\[\int \sec^4 x\ dx\]

Q 71 | Page 204

\[\int {cosec}^4 2x\ dx\]

Q 72 | Page 204

\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} dx\]

Q 73 | Page 204

\[\int\frac{1}{2 + \cos x} dx\]

Q 74 | Page 204
\[\int\sqrt{\frac{a + x}{x}}dx\]
 
Q 75 | Page 204
\[\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} dx\]
Q 76 | Page 204
\[\int\frac{\sin^5 x}{\cos^4 x} dx\]
Q 77 | Page 204
\[\int\frac{\cos^5 x}{\sin x} dx\]
Q 78 | Page 204
\[\int\frac{\sin^6 x}{\cos x} dx\]
Q 79 | Page 204
\[\int\frac{\sin^2 x}{\cos^6 x} dx\]
Q 80 | Page 204
\[\int \sec^6 x\ dx\]
Q 81 | Page 204
\[\int \tan^5 x\ \sec^3 x\ dx\]
Q 82 | Page 204
\[\int \tan^3 x\ \sec^4 x\ dx\]
Q 83 | Page 204
\[\int\frac{1}{\sec x + cosec x} dx\]
Q 84 | Page 204
\[\int\sqrt{a^2 + x^2} dx\]
Q 85 | Page 204
\[\int\sqrt{x^2 - a^2} dx\]
Q 86 | Page 204
\[\int\sqrt{a^2 - x^2} dx\]
Q 87 | Page 204
\[\int\sqrt{3 x^2 + 4x + 1} dx\]
Q 88 | Page 204
\[\int\sqrt{1 + 2x - 3 x^2} dx\]
Q 89 | Page 204
\[\int x\sqrt{1 + x - x^2} dx\]
Q 90 | Page 204
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} dx\]
Q 91 | Page 204

\[ \int\left( 1 + x^2 \right) \cdot \cos 2x \cdot dx\]

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

Pages 1 - 39

Q 1 | Page 16
\[\int\limits_4^9 \frac{1}{\sqrt{x}} dx\]
Q 2 | Page 16
\[\int\limits_{- 2}^3 \frac{1}{x + 7} dx\]
Q 3 | Page 16
\[\int\limits_0^{1/2} \frac{1}{\sqrt{1 - x^2}} dx\]
Q 4 | Page 16
\[\int\limits_0^1 \frac{1}{1 + x^2} dx\]
Q 5 | Page 16
\[\int\limits_2^3 \frac{x}{x^2 + 1} dx\]
Q 6 | Page 16
\[\int\limits_0^\infty \frac{1}{a^2 + b^2 x^2} dx\]
Q 7 | Page 16
\[\int\limits_{- 1}^1 \frac{1}{1 + x^2} dx\]
Q 8 | Page 16
\[\int\limits_0^\infty e^{- x} dx\]
Q 9 | Page 16
\[\int\limits_0^1 \frac{x}{x + 1} dx\]
Q 10 | Page 16
\[\int\limits_0^\pi/2 \left( \sin x + \cos x \right) dx\]
Q 11 | Page 16

\[\int\limits_{\pi/4}^{\pi/2} \cot x\ dx\]

Q 12 | Page 16
\[\int\limits_0^{\pi/4} \sec x dx\]
Q 13 | Page 16
\[\int\limits_{\pi/6}^{\pi/4} cosec\ x\ dx\]
Q 14 | Page 16
\[\int\limits_0^1 \frac{1 - x}{1 + x} dx\]
Q 15 | Page 16
\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\]
Q 16 | Page 16
\[\int\limits_{- \pi/4}^{\pi/4} \frac{1}{1 + \sin x} dx\]
Q 17 | Page 16
\[\int\limits_0^{\pi/2} \cos^2 x\ dx\]
Q 18 | Page 16
\[\int\limits_0^{\pi/2} \cos^3 x\ dx\]
Q 19 | Page 16
\[\int\limits_0^{\pi/6} \cos x \cos 2x\ dx\]
Q 20 | Page 16
\[\int\limits_0^{\pi/2} \sin x \sin 2x\ dx\]
Q 21 | Page 16
\[\int\limits_{\pi/3}^{\pi/4} \left( \tan x + \cot x \right)^2 dx\]
Q 22 | Page 16
\[\int\limits_0^{\pi/2} \cos^4\ x\ dx\]

 

Q 23 | Page 16
\[\int\limits_0^{\pi/2} \left( a^2 \cos^2 x + b^2 \sin^2 x \right) dx\]
Q 24 | Page 16
\[\int\limits_0^{\pi/2} \sqrt{1 + \sin x}\ dx\]
Q 25 | Page 16
\[\int\limits_0^{\pi/2} \sqrt{1 + \cos x}\ dx\]
Q 26 | Page 16

Evaluate the following definite integrals:

\[\int_0^\frac{\pi}{2} x^2 \sin\ x\ dx\]
Q 27 | Page 17
\[\int\limits_0^{\pi/2} x \cos\ x\ dx\]
Q 28 | Page 17
\[\int\limits_0^{\pi/2} x^2 \cos\ x\ dx\]
Q 29 | Page 17
\[\int\limits_0^{\pi/4} x^2 \sin\ x\ dx\]
Q 29 | Page 39
\[\int\limits_0^{\pi/2} \frac{x + \sin x}{1 + \cos x} dx\]
Q 30 | Page 17
\[\int\limits_0^{\pi/2} x^2 \cos\ 2x\ dx\]
Q 31 | Page 17
\[\int\limits_0^{\pi/2} x^2 \cos^2 x\ dx\]
Q 32 | Page 17
\[\int\limits_1^2 \log\ x\ dx\]
Q 33 | Page 17
\[\int\limits_1^3 \frac{\log x}{\left( x + 1 \right)^2} dx\]
Q 34 | Page 17
\[\int\limits_1^e \frac{e^x}{x} \left( 1 + x \log x \right) dx\]
Q 35 | Page 17
\[\int\limits_1^e \frac{\log x}{x} dx\]
Q 36 | Page 17
\[\int\limits_e^{e^2} \left\{ \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right\} dx\]
Q 37 | Page 17
\[\int\limits_1^2 \frac{x + 3}{x \left( x + 2 \right)} dx\]
Q 38 | Page 17
\[\int\limits_0^1 \frac{2x + 3}{5 x^2 + 1} dx\]
Q 39 | Page 17
\[\int\limits_0^2 \frac{1}{4 + x - x^2} dx\]
Q 40 | Page 17
\[\int\limits_0^1 \frac{1}{2 x^2 + x + 1} dx\]
Q 41 | Page 17
\[\int\limits_0^1 \sqrt{x \left( 1 - x \right)} dx\]
Q 42 | Page 17
\[\int\limits_0^2 \frac{1}{\sqrt{3 + 2x - x^2}} dx\]
Q 43 | Page 17
\[\int\limits_0^4 \frac{1}{\sqrt{4x - x^2}} dx\]
Q 44 | Page 17
\[\int\limits_{- 1}^1 \frac{1}{x^2 + 2x + 5} dx\]
Q 45 | Page 17
\[\int\limits_1^4 \frac{x^2 + x}{\sqrt{2x + 1}} dx\]
Q 46 | Page 17
\[\int\limits_0^1 x \left( 1 - x \right)^5 dx\]
Q 47 | Page 17
\[\int\limits_1^2 \left( \frac{x - 1}{x^2} \right) e^x dx\]
Q 48 | Page 17
\[\int\limits_0^1 \left( x e^{2x} + \sin\frac{\ pix}{2} \right) dx\]
Q 49 | Page 17

\[\int\limits_0^1 \left( x e^x + \cos\frac{\ pix}{4} \right) dx\]

 

Q 50 | Page 17
\[\int\limits_{\pi/2}^\pi e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
Q 51 | Page 17
\[\int\limits_0^{2\pi} e^{x/2} \sin\left( \frac{x}{2} + \frac{\pi}{4} \right) dx\]
Q 52 | Page 17
\[\int\limits_0^{2\pi} e^x \cos\left( \frac{\pi}{4} + \frac{x}{2} \right) dx\]
Q 53 | Page 17
\[\int_0^\pi e^{2x} \cdot \sin\left( \frac{\pi}{4} + x \right) dx\]
Q 54 | Page 17
\[\int\limits_0^1 \frac{1}{\sqrt{1 + x} - \sqrt{x}} dx\]
Q 55 | Page 17
\[\int\limits_1^2 \frac{x}{\left( x + 1 \right) \left( x + 2 \right)} dx\]
Q 56 | Page 17
\[\int\limits_0^{\pi/2} \sin^3 x\ dx\]
Q 57 | Page 17
\[\int\limits_0^\pi \left( \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} \right) dx\]
Q 58 | Page 17
\[\int\limits_1^2 e^{2x} \left( \frac{1}{x} - \frac{1}{2 x^2} \right) dx\]
Q 59 | Page 17

Evaluate the following definite integral:

\[\int_0^1 \frac{1}{\sqrt{\left( x - 1 \right)\left( 2 - x \right)}}dx\]
Q 60 | Page 17

\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.

Q 61 | Page 18

\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.

Q 62 | Page 18
\[\int_\pi^\frac{3\pi}{2} \sqrt{1 - \cos2x}dx\]
Q 63 | Page 18
\[\int_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\]
Q 64 | Page 18
\[\int_0^\frac{\pi}{4} \left( \tan x + \cot x \right)^{- 2} dx\]
Q 65 | Page 18
\[\int_0^1 x\log\left( 1 + 2x \right)dx\]
Q 66 | Page 18
\[\int_\frac{\pi}{6}^\frac{\pi}{3} \left( \tan x + \cot x \right)^2 dx\]
Q 67 | Page 1
\[\int_0^\frac{\pi}{4} \left( a^2 \cos^2 x + b^2 \sin^2 x \right)dx\]
Q 68 | Page 18
\[\int_0^1 \frac{1}{1 + 2x + 2 x^2 + 2 x^3 + x^4}dx\]

Pages 38 - 40

Q 1 | Page 38
\[\int\limits_2^4 \frac{x}{x^2 + 1} dx\]
Q 2 | Page 38
\[\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx\]
Q 3 | Page 38
\[\int\limits_1^2 \frac{3x}{9 x^2 - 1} dx\]
Q 4 | Page 38
\[\int\limits_0^{\pi/2} \frac{1}{5 \cos x + 3 \sin x} dx\]
Q 5 | Page 38
\[\int\limits_0^a \frac{x}{\sqrt{a^2 + x^2}} dx\]
Q 6 | Page 38
\[\int\limits_0^1 \frac{e^x}{1 + e^{2x}} dx\]
Q 7 | Page 38
\[\int\limits_0^1 x e^{x^2} dx\]
Q 8 | Page 38
\[\int\limits_1^3 \frac{\cos \left( \log x \right)}{x} dx\]
Q 9 | Page 38
\[\int\limits_0^1 \frac{2x}{1 + x^4} dx\]
Q 10 | Page 38
\[\int\limits_0^a \sqrt{a^2 - x^2} dx\]
Q 11 | Page 39
\[\int\limits_0^{\pi/2} \sqrt{\sin \phi} \cos^5 \phi\ d\phi\]

 

Q 12 | Page 39
\[\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx\]
Q 13 | Page 39
\[\int\limits_0^{\pi/2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d\theta\]
Q 14 | Page 39
\[\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx\]
Q 15 | Page 39
\[\int\limits_0^1 \frac{\sqrt{\tan^{- 1} x}}{1 + x^2} dx\]
Q 16 | Page 39
\[\int\limits_0^2 x\sqrt{x + 2}\ dx\]
Q 17 | Page 39
\[\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx\]
Q 18 | Page 39
\[\int\limits_0^{\pi/2} \frac{\sin x \cos x}{1 + \sin^4 x} dx\]
Q 19 | Page 39
\[\int\limits_0^{\pi/2} \frac{dx}{a \cos x + b \sin x}a, b > 0\]
Q 20 | Page 39
\[\int\limits_0^{\pi/2} \frac{1}{5 + 4 \sin x} dx\]
Q 21 | Page 39
\[\int\limits_0^\pi \frac{\sin x}{\sin x + \cos x} dx\]
Q 22 | Page 39
\[\int\limits_0^\pi \frac{1}{3 + 2 \sin x + \cos x} dx\]
Q 23 | Page 39
\[\int\limits_0^1 \tan^{- 1} x\ dx\]
Q 24 | Page 39
\[\int_0^\frac{1}{2} \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx\]
Q 25 | Page 39
\[\int\limits_0^{\pi/4} \left( \sqrt{\tan}x + \sqrt{\cot}x \right) dx\]
Q 26 | Page 39
\[\int\limits_0^{\pi/4} \frac{\tan^3 x}{1 + \cos 2x} dx\]
Q 27 | Page 39
\[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\]
Q 28 | Page 39
\[\int\limits_0^{\pi/2} \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx\]
Q 30 | Page 39
\[\int\limits_0^1 \frac{\tan^{- 1} x}{1 + x^2} dx\]
Q 31 | Page 39
\[\int_0^\frac{\pi}{4} \frac{\sin x + \cos x}{3 + \sin2x}dx\]
Q 32 | Page 39
\[\int\limits_0^1 x \tan^{- 1} x\ dx\]
Q 33 | Page 39
\[\int\limits_0^1 \frac{1 - x^2}{x^4 + x^2 + 1} dx\]
Q 34 | Page 39
\[\int\limits_0^1 \frac{24 x^3}{\left( 1 + x^2 \right)^4} dx\]
Q 35 | Page 39
\[\int\limits_4^{12} x \left( x - 4 \right)^{1/3} dx\]
Q 36 | Page 39
\[\int\limits_0^{\pi/2} x^2 \sin\ x\ dx\]
Q 37 | Page 39
\[\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx\]
Q 38 | Page 39
\[\int\limits_0^1 \frac{1 - x^2}{\left( 1 + x^2 \right)^2} dx\]
Q 39 | Page 39
\[\int\limits_{- 1}^1 5 x^4 \sqrt{x^5 + 1} dx\]
Q 40 | Page 39
\[\int_0^\frac{\pi}{2} \frac{\cos^2 x}{1 + 3 \sin^2 x}dx\]
Q 41 | Page 39
\[\int\limits_0^{\pi/4} \sin^3 2t \cos 2t\ dt\]
Q 42 | Page 39
\[\int\limits_0^\pi 5 \left( 5 - 4 \cos \theta \right)^{1/4} \sin \theta\ d \theta\]
Q 43 | Page 39
\[\int\limits_0^{\pi/6} \cos^{- 3} 2 \theta \sin 2\ \theta\ d\ \theta\]
Q 44 | Page 39

\[\int\limits_0^{( \pi )^{2/3}} \sqrt{x} \cos^2 x^{3/2} dx\]

Q 45 | Page 40
\[\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx\]
Q 46 | Page 40
\[\int\limits_0^{\pi/2} \cos^5 x\ dx\]
Q 47 | Page 40
\[\int\limits_4^9 \frac{\sqrt{x}}{\left( 30 - x^{3/2} \right)^2} dx\]
Q 48 | Page 40
\[\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx\]
Q 49 | Page 40
\[\int\limits_0^{\pi/2} 2 \sin x \cos x \tan^{- 1} \left( \sin x \right) dx\]
Q 50 | Page 39
\[\int\limits_0^{\pi/2} \sin 2x \tan^{- 1} \left( \sin x \right) dx\]
Q 51 | Page 40
\[\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx\]
Q 52 | Page 40
\[\int\limits_0^a \sin^{- 1} \sqrt{\frac{x}{a + x}} dx\]
Q 53 | Page 40
\[\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{3/2}} dx\]
Q 54 | Page 40
\[\int\limits_0^a x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} dx\]
Q 55 | Page 40
\[\int\limits_{- a}^a \sqrt{\frac{a - x}{a + x}} dx\]
Q 56 | Page 40
\[\int\limits_0^{\pi/2} \frac{\sin x \cos x}{\cos^2 x + 3 \cos x + 2} dx\]
Q 57 | Page 40
\[\int_0^\frac{\pi}{2} \frac{\tan x}{1 + m^2 \tan^2 x}dx\]
Q 58 | Page 40
\[\int_0^\frac{1}{2} \frac{1}{\left( 1 + x^2 \right)\sqrt{1 - x^2}}dx\]
Q 59 | Page 40
\[\int_\frac{1}{3}^1 \frac{\left( x - x^3 \right)^\frac{1}{3}}{x^4}dx\]
Q 60 | Page 40
\[\int_0^\frac{\pi}{4} \frac{\sin^2 x \cos^2 x}{\left( \sin^3 x + \cos^3 x \right)^2}dx\]
Q 61 | Page 40
\[\int_0^\frac{\pi}{2} \sqrt{\cos x - \cos^3 x}\left( \sec^2 x - 1 \right) \cos^2 xdx\]
Q 62 | Page 40
\[\int_0^\frac{\pi}{2} \frac{\cos x}{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^n}dx\]

Page 55

Q 1.1 | Page 55
\[\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \binom{4x + 3, if 1 \leq x \leq 2}{3x + 5, if 2 \leq x \leq 4}\]

 

Q 1.2 | Page 55
\[\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \binom{4x + 3, if 1 \leq x \leq 2}{3x + 5, if 2 \leq x \leq 4}\]

 

Q 1.3 | Page 55

\[\int\limits_1^4 f\left( x \right) dx\, where\ f\left( x \right) = \begin{cases}7x + 3 & , & if 1 \leq x \leq 3 \\ 8x & , & if 3 \leq x \leq 4\end{cases}\]

Pages 56 - 61

Q 2 | Page 56

Evaluate the following integral:

\[\int\limits_{- 4}^4 \left| x + 2 \right| dx\]
Q 3 | Page 56

Evaluate the following integral:

\[\int\limits_{- 3}^3 \left| x + 1 \right| dx\]
Q 4 | Page 61

Evaluate each of the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}dx\]

 

Q 4 | Page 56

Evaluate the following integral:

\[\int\limits_{- 1}^1 \left| 2x + 1 \right| dx\]
Q 5 | Page 56

Evaluate the following integral:

\[\int\limits_{- 2}^2 \left| 2x + 3 \right| dx\]
Q 6 | Page 56

Evaluate the following integral:

\[\int\limits_0^2 \left| x^2 - 3x + 2 \right| dx\]

 

Q 7 | Page 56

Evaluate the following integral:

\[\int\limits_0^3 \left| 3x - 1 \right| dx\]

 

Q 8 | Page 56

Evaluate the following integral:

\[\int\limits_{- 6}^6 \left| x + 2 \right| dx\]

 

Q 9 | Page 56

Evaluate the following integral:

\[\int\limits_{- 2}^2 \left| x + 1 \right| dx\]

 

Q 10 | Page 56

Evaluate the following integral:

\[\int\limits_1^2 \left| x - 3 \right| dx\]

 

Q 11 | Page 56

Evaluate the following integral:

\[\int\limits_0^\pi/2 \left| \cos 2x \right| dx\]
Q 12 | Page 56

Evaluate the following integral:

\[\int\limits_0^{2\pi} \left| \sin x \right| dx\]

 

Q 13 | Page 56

Evaluate the following integral:

\[\int\limits_{- \pi/4}^{\pi/4} \left| \sin x \right| dx\]
Q 14 | Page 56

Evaluate the following integral:

\[\int\limits_2^8 \left| x - 5 \right| dx\]

 

Q 15 | Page 56

Evaluate the following integral:

\[\int\limits_{- \pi/2}^{\pi/2} \left\{ \sin \left| x \right| + \cos \left| x \right| \right\} dx\]

 

Q 16 | Page 56

Evaluate the following integral:

\[\int\limits_0^4 \left| x - 1 \right| dx\]
Q 17 | Page 56

Evaluate the following integral:

\[\int\limits_1^4 \left\{ \left| x - 1 \right| + \left| x - 2 \right| + \left| x - 4 \right| \right\} dx\]

 

Q 18 | Page 56

Evaluate the following integral:

\[\int\limits_{- 5}^0 f\left( x \right) dx, where\ f\left( x \right) = \left| x \right| + \left| x + 2 \right| + \left| x + 5 \right|\]

 

Q 19 | Page 56

Evaluate the following integral:

\[\int\limits_0^4 \left( \left| x \right| + \left| x - 2 \right| + \left| x - 4 \right| \right) dx\]
Q 20 | Page 56
\[\int_{- 1}^2 \left( \left| x + 1 \right| + \left| x \right| + \left| x - 1 \right| \right)dx\]

 

Q 21 | Page 56
\[\int_{- 2}^2 x e^\left| x \right| dx\]
Q 22 | Page 56
\[\int_{- \frac{\pi}{4}}^\frac{\pi}{2} \sin x\left| \sin x \right|dx\]

 

Q 23 | Page 56
\[\int_0^\pi \cos x\left| \cos x \right|dx\]
Q 24 | Page 56
\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left( 2\sin\left| x \right| + \cos\left| x \right| \right)dx\]
Q 25 | Page 56
\[\int_{- \frac{\pi}{2}}^\pi \sin^{- 1} \left( \sin x \right)dx\]
Q 26 | Page 56
\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{- \frac{\pi}{2}}{\sqrt{\cos x \sin^2 x}}dx\]
Q 27 | Page 56
\[\int_0^2 2x\left[ x \right]dx\]
Q 28 | Page 56
\[\int_0^{2\pi} \cos^{- 1} \left( \cos x \right)dx\]

Page 61

Q 1 | Page 61

Evaluate each of the following integral:

\[\int_0^{2\pi} \frac{e^\ sin x}{e^\ sin x + e^{- \ sin x}}dx\]

 

Q 2 | Page 61

Evaluate each of the following integral:

\[\int_0^{2\pi} \log\left( \sec x + \tan x \right)dx\]

 

Q 3 | Page 61

Evaluate each of the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}}dx\]
Q 5 | Page 61

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{\tan^2 x}{1 + e^x}dx\]

 

Q 6 | Page 61

Evaluate each of the following integral:

\[\int_{- a}^a \frac{1}{1 + a^x}dx\]
Q 7 | Page 61

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{3}}^\frac{\pi}{3} \frac{1}{1 + e^\ tan\ x}dx\]

 

Q 8 | Page 61

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{\cos^2 x}{1 + e^x}dx\]
Q 9 | Page 61

Evaluate each of the following integral:

\[\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{x^{11} - 3 x^9 + 5 x^7 - x^5 + 1}{\cos^2 x}dx\]
Q 10 | Page 61

Evaluate each of the following integral:

\[\int_a^b \frac{x^\frac{1}{n}}{x^\frac{1}{n} + \left( a + b - x \right)^\frac{1}{n}}dx, n \in N, n \geq 2\]

Q 11 | Page 61
\[\int\limits_0^{\pi/2} \left( 2 \log \cos x - \log \sin 2x \right) dx\]

 

Q 12 | Page 61
\[\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx\]
Q 13 | Page 61
\[\int\limits_0^5 \frac{\sqrt[4]{x + 4}}{\sqrt[4]{x + 4} + \sqrt[4]{9 - x}} dx\]
Q 14 | Page 61
\[\int\limits_0^7 \frac{\sqrt[3]{x}}{\sqrt[3]{x} + \sqrt[3]{7} - x} dx\]
Q 15 | Page 61
\[\int\limits_{\pi/6}^{\pi/3} \frac{1}{1 + \sqrt{\tan x}} dx\]
Q 16 | Page 61

If  \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

\[\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx\]

 

Pages 94 - 96

Q 1 | Page 94
\[\int\limits_0^{\pi/2} \frac{dx}{1 + \tan x}\]

 

Q 2 | Page 94
\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot x} dx\]
Q 3 | Page 94
\[\int\limits_0^{\pi/2} \frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} dx\]
Q 4 | Page 94
\[\int\limits_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx\]
Q 5 | Page 94
\[\int\limits_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx\]

 

Q 6 | Page 94
\[\int\limits_0^{\pi/2} \frac{1}{1 + \sqrt{\tan x}} dx\]
Q 7 | Page 95
\[\int\limits_0^a \frac{1}{x + \sqrt{a^2 - x^2}} dx\]
Q 8 | Page 95
\[\int\limits_0^\infty \frac{\log x}{1 + x^2} dx\]
Q 9 | Page 95
\[\int\limits_0^1 \frac{\log\left( 1 + x \right)}{1 + x^2} dx\]

 

Q 10 | Page 95
\[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]
Q 11 | Page 95
\[\int\limits_0^\pi \frac{x \tan x}{\sec x cosec x} dx\]
Q 12 | Page 95
\[\int\limits_0^\pi x \sin x \cos^4 x\ dx\]
Q 13 | Page 95
\[\int\limits_0^\pi x \sin^3 x\ dx\]
Q 14 | Page 95
\[\int\limits_0^\pi x \log \sin x\ dx\]
Q 15 | Page 95
\[\int\limits_0^\pi \frac{x \sin x}{1 + \sin x} dx\]
Q 16 | Page 95
\[\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx, 0 < \alpha < \pi\]
Q 17 | Page 95
\[\int\limits_0^\pi x \cos^2 x\ dx\]
Q 18 | Page 95

Evaluate the following integral:

\[\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{1}{1 + \cot^\frac{3}{2} x}dx\]

 

Q 19 | Page 95

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx\]
Q 20 | Page 95

Evaluate the following integral:

\[\int_2^8 \frac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}dx\]
Q 21 | Page 95

Evaluate the following integral:

\[\int_0^\pi x\sin x \cos^2 xdx\]
Q 22 | Page 95
\[\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx\]
Q 23 | Page 95
\[\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx\]
Q 24 | Page 95
\[\int\limits_{- \pi/2}^{\pi/2} \sin^4 x\ dx\]
Q 25 | Page 95
\[\int\limits_{- 1}^1 \log\left( \frac{2 - x}{2 + x} \right) dx\]
Q 26 | Page 95
\[\int\limits_{- \pi/4}^{\pi/4} \sin^2 x\ dx\]
Q 27 | Page 95
\[\int\limits_0^\pi \log\left( 1 - \cos x \right) dx\]
Q 28 | Page 95
\[\int\limits_{- \pi/2}^{\pi/2} \log\left( \frac{2 - \sin x}{2 + \sin x} \right) dx\]
Q 29 | Page 95

Evaluate the following integral:

\[\int_{- \pi}^\pi \frac{2x\left( 1 + \sin x \right)}{1 + \cos^2 x}dx\]
Q 30 | Page 95

Evaluate the following integral:

\[\int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta\]
Q 31 | Page 95

Evaluate the following integral:

\[\int_{- 2}^2 \frac{3 x^3 + 2\left| x \right| + 1}{x^2 + \left| x \right| + 1}dx\]
Q 32 | Page 95

Evaluate the following integral:

\[\int_{- \frac{3\pi}{2}}^{- \frac{\pi}{2}} \left\{ \sin^2 \left( 3\pi + x \right) + \left( \pi + x \right)^3 \right\}dx\]
Q 33 | Page 95
\[\int\limits_0^2 x\sqrt{2 - x} dx\]
Q 34 | Page 95
\[\int\limits_0^1 \log\left( \frac{1}{x} - 1 \right) dx\]

 

Q 35 | Page 95

Evaluate the following integral:

\[\int_{- 1}^1 \left| xcos\pi x \right|dx\]

 

Q 36 | Page 95

Evaluate the following integral:

\[\int_0^\pi \left( \frac{x}{1 + \sin^2 x} + \cos^7 x \right)dx\]
Q 37 | Page 95

Evaluate 

\[\int\limits_0^\pi \frac{x}{1 + \sin \alpha \sin x}dx\]

Q 38 | Page 95

Evaluate the following integral:

\[\int_0^{2\pi} \sin^{100} x \cos^{101} xdx\]

 

Q 39 | Page 95

Evaluate the following integral:

\[\int_0^\frac{\pi}{2} \frac{a\sin x + b\sin x}{\sin x + \cos x}dx\]

 

Q 41 | Page 95
\[\int_0^1 | x\sin \pi x | dx\]
Q 42 | Page 95

Evaluate : 

\[\int\limits_0^{3/2} \left| x \sin \pi x \right|dx\]
Q 43 | Page 96

If f is an integrable function such that f(2a − x) = f(x), then prove that

\[\int\limits_0^{2a} f\left( x \right) dx = 2 \int\limits_0^a f\left( x \right) dx\]

 

Q 44 | Page 96

If f(2a − x) = −f(x), prove that

\[\int\limits_0^{2a} f\left( x \right) dx = 0 .\]
Q 45.1 | Page 96

If f is an integrable function, show that

(i)

\[\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx\]
Q 45.2 | Page 96

If f is an integrable function, show that

\[\int\limits_{- a}^a x f\left( x^2 \right) dx = 0\]

 

Q 46 | Page 96

If f (x) is a continuous function defined on [0, 2a]. Then, prove that

\[\int\limits_0^{2a} f\left( x \right) dx = \int\limits_0^a \left\{ f\left( x \right) + f\left( 2a - x \right) \right\} dx\]

 

Q 47 | Page 96

If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

\[\int_a^b xf\left( x \right)dx = \left( \frac{a + b}{2} \right) \int_a^b f\left( x \right)dx\]
Q 48 | Page 96

If f(x) is a continuous function defined on [−aa], then prove that 

\[\int\limits_{- a}^a f\left( x \right) dx = \int\limits_0^a \left\{ f\left( x \right) + f\left( - x \right) \right\} dx\]
Q 49 | Page 96

Prove that:

\[\int_0^\pi xf\left( \sin x \right)dx = \frac{\pi}{2} \int_0^\pi f\left( \sin x \right)dx\]

Pages 110 - 111

Q 1 | Page 110
\[\int\limits_0^3 \left( x + 4 \right) dx\]
Q 2 | Page 110
\[\int\limits_0^2 \left( x + 3 \right) dx\]
Q 3 | Page 110
\[\int\limits_1^3 \left( 3x - 2 \right) dx\]
Q 4 | Page 110
\[\int\limits_{- 1}^1 \left( x + 3 \right) dx\]
Q 5 | Page 110
\[\int\limits_0^5 \left( x + 1 \right) dx\]
Q 6 | Page 110
\[\int\limits_1^3 \left( 2x + 3 \right) dx\]
Q 7 | Page 110
\[\int\limits_3^5 \left( 2 - x \right) dx\]
Q 8 | Page 110
\[\int\limits_0^2 \left( x^2 + 1 \right) dx\]
Q 9 | Page 110
\[\int\limits_1^2 x^2 dx\]
Q 10 | Page 110
\[\int\limits_2^3 \left( 2 x^2 + 1 \right) dx\]
Q 11 | Page 110
\[\int\limits_1^2 \left( x^2 - 1 \right) dx\]
Q 12 | Page 110
\[\int\limits_0^2 \left( x^2 + 4 \right) dx\]
Q 13 | Page 111
\[\int\limits_1^4 \left( x^2 - x \right) dx\]
Q 14 | Page 111
\[\int\limits_0^1 \left( 3 x^2 + 5x \right) dx\]
Q 15 | Page 111
\[\int\limits_0^2 e^x dx\]
Q 16 | Page 111
\[\int\limits_a^b e^x dx\]
Q 17 | Page 111
\[\int\limits_a^b \cos\ x\ dx\]
Q 18 | Page 111
\[\int\limits_0^{\pi/2} \sin x\ dx\]
Q 19 | Page 111
\[\int\limits_0^{\pi/2} \cos x\ dx\]
Q 20 | Page 111
\[\int\limits_1^4 \left( 3 x^2 + 2x \right) dx\]
Q 21 | Page 111
\[\int\limits_0^2 \left( 3 x^2 - 2 \right) dx\]
Q 22 | Page 111
\[\int\limits_0^2 \left( x^2 + 2 \right) dx\]
Q 23 | Page 111
\[\int\limits_0^2 \left( x^2 + 2x + 1 \right) dx\]
Q 23 | Page 111
\[\int\limits_0^4 \left( x + e^{2x} \right) dx\]
Q 24 | Page 111
\[\int\limits_0^2 \left( x^2 + x \right) dx\]
Q 25 | Page 111
\[\int\limits_0^2 \left( x^2 + 2x + 1 \right) dx\]
Q 26 | Page 111
\[\int\limits_0^3 \left( 2 x^2 + 3x + 5 \right) dx\]
Q 27 | Page 111
\[\int\limits_a^b x\ dx\]
Q 28 | Page 111
\[\int\limits_0^5 \left( x + 1 \right) dx\]
Q 29 | Page 111
\[\int\limits_2^3 x^2 dx\]
Q 30 | Page 111
\[\int\limits_1^4 \left( x^2 - x \right) dx\]
Q 31 | Page 111
\[\int\limits_0^2 \left( x^2 - x \right) dx\]
Q 32 | Page 111
\[\int\limits_1^3 \left( 2 x^2 + 5x \right) dx\]
Q 33 | Page 111

Evaluate the following integrals as limit of sums:

\[\int_1^3 \left( 3 x^2 + 1 \right)dx\]

Pages 111 - 116

Q 1 | Page 115
\[\int\limits_0^{\pi/2} \sin^2 x\ dx .\]
Q 2 | Page 115
\[\int\limits_0^{\pi/2} \cos^2 x\ dx .\]
Q 3 | Page 115
\[\int\limits_{- \pi/2}^{\pi/2} \sin^2 x\ dx .\]
Q 4 | Page 115
\[\int\limits_{- \pi/2}^{\pi/2} \cos^2 x\ dx .\]
Q 5 | Page 111
\[\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx .\]
Q 6 | Page 115
\[\int\limits_{- \pi/2}^{\pi/2} x \cos^2 x\ dx .\]

 

Q 7 | Page 115
\[\int\limits_0^{\pi/4} \tan^2 x\ dx .\]
Q 8 | Page 115
\[\int\limits_0^1 \frac{1}{x^2 + 1} dx\]
Q 9 | Page 115
\[\int\limits_{- 2}^1 \frac{\left| x \right|}{x} dx .\]
Q 10 | Page 115
\[\int\limits_0^\infty e^{- x} dx .\]
Q 11 | Page 115
\[\int\limits_0^4 \frac{1}{\sqrt{16 - x^2}} dx .\]
Q 12 | Page 115
\[\int\limits_0^3 \frac{1}{x^2 + 9} dx .\]
Q 13 | Page 115
\[\int\limits_0^{\pi/2} \sqrt{1 - \cos 2x}\ dx .\]
Q 14 | Page 115
\[\int\limits_0^{\pi/2} \log \tan x\ dx .\]
Q 15 | Page 115
\[\int\limits_0^{\pi/2} \log \left( \frac{3 + 5 \cos x}{3 + 5 \sin x} \right) dx .\]

 

Q 16 | Page 115
\[\int\limits_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx, n \in N .\]
Q 17 | Page 115
\[\int\limits_0^\pi \cos^5 x\ dx .\]
Q 18 | Page 115
\[\int\limits_{- \pi/2}^{\pi/2} \log\left( \frac{a - \sin \theta}{a + \sin \theta} \right) d\theta\]
Q 19 | Page 115
\[\int\limits_{- 1}^1 x\left| x \right| dx .\]
Q 20 | Page 115
\[\int\limits_a^b \frac{f\left( x \right)}{f\left( x \right) + f\left( a + b - x \right)} dx .\]
Q 21 | Page 115
\[\int\limits_0^1 \frac{1}{1 + x^2} dx\]
Q 22 | Page 115

Evaluate each of the following integral:

\[\int_0^\frac{\pi}{4} \tan\ xdx\]

 

Q 23 | Page 115
\[\int\limits_2^3 \frac{1}{x}dx\]
Q 24 | Page 115
\[\int\limits_0^2 \sqrt{4 - x^2} dx\]
Q 25 | Page 115
\[\int\limits_0^1 \frac{2x}{1 + x^2} dx\]
Q 26 | Page 115

Evaluate each of the following  integral:

\[\int_0^1 x e^{x^2} dx\]

 

Q 27 | Page 115

Evaluate each of the following integral:

\[\int_0^\frac{\pi}{4} \sin2xdx\]
Q 28 | Page 115

Evaluate each of the following integral:

\[\int_e^{e^2} \frac{1}{x\log x}dx\]
Q 29 | Page 115

Evaluate each of the following integral:

\[\int_0^\frac{\pi}{2} e^x \left( \sin x - \cos x \right)dx\]

 

Q 30 | Page 115

Solve each of the following integral:

\[\int_2^4 \frac{x}{x^2 + 1}dx\]
Q 31 | Page 116

If \[\int\limits_0^1 \left( 3 x^2 + 2x + k \right) dx = 0,\] find the value of k.

 

Q 32 | Page 116

If \[\int\limits_0^a 3 x^2 dx = 8,\] write the value of a.

 

 

Q 33 | Page 116

If \[f\left( x \right) = \int_0^x t\sin tdt\], the write the value of \[f'\left( x \right)\]

Q 34 | Page 116

If \[\int_0^a \frac{1}{4 + x^2}dx = \frac{\pi}{8}\] , find the value of a.

Q 35 | Page 116

Write the coefficient abc of which the value of the integral

\[\int\limits_{- 3}^3 \left( a x^2 + bx + c \right) dx\] is independent.
Q 36 | Page 116

Evaluate : 

\[\int\limits_2^3 3^x dx .\]
Q 37 | Page 116
\[\int\limits_0^2 \left[ x \right] dx .\]
Q 38 | Page 116
\[\int\limits_0^{15} \left[ x \right] dx .\]
Q 39 | Page 116

\[\int\limits_0^1 \left\{ x \right\} dx,\] where {x} denotes the fractional part of x.  

 
Q 40 | Page 116
\[\int\limits_0^1 e^\left\{ x \right\} dx .\]
Q 41 | Page 116
\[\int\limits_0^2 x\left[ x \right] dx .\]
Q 42 | Page 116
\[\int\limits_0^1 2^{x - \left[ x \right]} dx\]
Q 43 | Page 116
\[\int\limits_1^2 \log_e \left[ x \right] dx .\]
Q 44 | Page 116
\[\int\limits_0^\sqrt{2} \left[ x^2 \right] dx .\]
Q 45 | Page 116

If \[\left[ \cdot \right] and \left\{ \cdot \right\}\] denote respectively the greatest integer and fractional part functions respectively, evaluate the following integrals:

\[\int\limits_0^\pi/4 \sin \left\{ x \right\} dx\]

 

Pages 117 - 120

Q 1 | Page 117
\[\int\limits_0^1 \sqrt{x \left( 1 - x \right)} dx\] equals

π/2

π/4

π/6

π/8

Q 2 | Page 117

\[\int\limits_0^\pi \frac{1}{1 + \sin x} dx\] equals

0

1/2

2

3/2

Q 3 | Page 117

\[\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx\] is 

\[\frac{\pi^2}{4}\]
\[\frac{\pi^2}{2}\]
\[\frac{3 \pi^2}{2}\]

\[\frac{\pi^2}{3}\]

Q 4 | Page 117

The value of \[\int\limits_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx\] is 

0

2

8

4

Q 5 | Page 117

The value of the integral \[\int\limits_0^\pi/2 \frac{\sqrt{\cos x}}{\sqrt{\cos x} + \sqrt{\sin x}} dx\]  is 

0

π/2

π/4

none of these

Q 6 | Page 117

\[\int\limits_0^\infty \frac{1}{1 + e^x} dx\]  equals

 log 2 − 1

 log 2

log 4 − 1

 − log 2

Q 7 | Page 117

\[\int\limits_0^{\pi^2 /4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx\]  equals

2

1

π/4

π2/8

Q 8 | Page 117
\[\int\limits_0^\pi/2 \frac{\cos x}{\left( 2 + \sin x \right)\left( 1 + \sin x \right)} dx\] equals
\[\log\left( \frac{2}{3} \right)\]
\[\log\left( \frac{3}{2} \right)\]
\[\log\left( \frac{3}{4} \right)\]
\[\log\left( \frac{4}{3} \right)\]
Q 9 | Page 117

\[\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx\]

\[\frac{1}{3} \tan^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]
\[\frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{1}{\sqrt{3}} \right)\]
\[\sqrt{3} \tan^{- 1} \left( \sqrt{3} \right)\]

 

\[2\sqrt{3} \tan^{- 1} \sqrt{3}\]
Q 10 | Page 117

\[\int\limits_0^\pi \sqrt{\frac{1 - x}{1 + x}}dx =\] 

\[\frac{\pi}{2}\]

\[\frac{\pi}{2} - 1\]

\[\frac{\pi}{2} + 1\]

 π + 1

non of above this

Q 11 | Page 117
\[\int\limits_0^\pi \frac{1}{a + b \cos x} dx =\]

\[\frac{\pi}{\sqrt{a^2 - b^2}}\]

\[\frac{\pi}{ab}\]

 

\[\frac{\pi}{a^2 + b^2}\]

(a + b) π

Q 12 | Page 118
\[\int\limits_{\pi/6}^{\pi/3} \frac{1}{1 + \sqrt{\cot}x} dx\] is

 π/3

 π/6

π/12

π/2

Q 13 | Page 118

Given that

\[\int\limits_0^\infty \frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)\left( x^2 + c^2 \right)} dx = \frac{\pi}{2\left( a + b \right)\left( b + c \right)\left( c + a \right)},\]  the value of

\[\int\limits_0^\infty \frac{dx}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)},\]
\[\frac{\pi}{60}\]
\[\frac{\pi}{20}\]
\[\frac{\pi}{40}\]
\[\frac{\pi}{80}\]
Q 14 | Page 118
\[\int\limits_1^e \log x\ dx =\]

1

 e − 1

e + 1

 0

Q 15 | Page 118
\[\int\limits_1^\sqrt{3} \frac{1}{1 + x^2} dx\]  is equal to
\[\frac{\pi}{12}\]
\[\frac{\pi}{6}\]
\[\frac{\pi}{4}\]
\[\frac{\pi}{3}\]
Q 16 | Page 118
\[\int\limits_0^3 \frac{3x + 1}{x^2 + 9} dx =\]
\[\frac{\pi}{12} + \log\left( 2\sqrt{2} \right)\]
\[\frac{\pi}{2} + \log\left( 2\sqrt{2} \right)\]
\[\frac{\pi}{3} + \log\left( 2\sqrt{2} \right)\]
Q 17 | Page 118

The value of the integral \[\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx\]

 

\[\frac{\pi}{2}\]
\[\frac{\pi}{4}\]
\[\frac{\pi}{6}\]
\[\frac{\pi}{3}\]
Q 18 | Page 118
\[\int\limits_{- \pi/2}^{\pi/2} \sin\left| x \right| dx\]  is equal to

 1

2

− 1

− 2

Q 19 | Page 118
\[\int\limits_0^{\pi/2} \frac{1}{1 + \tan x} dx\]  is equal to
\[\frac{ \pi}{4}\]
\[\frac{\pi}{3}\]
\[\frac{\pi}{2}\]

 π

Q 20 | Page 118

The value of \[\int\limits_0^{\pi/2} \cos x\ e^{\sin x}\ dx\] is

 

 1

e − 1

0

− 1 

Q 21 | Page 118

If \[\int\limits_0^a \frac{1}{1 + 4 x^2} dx = \frac{\pi}{8},\] then a equals

 

\[\frac{\pi}{2}\]
\[\frac{1}{2}\]
\[\frac{\pi}{4}\]

1

Q 22 | Page 118

If \[\int\limits_0^1 f\left( x \right) dx = 1, \int\limits_0^1 xf\left( x \right) dx = a, \int\limits_0^1 x^2 f\left( x \right) dx = a^2 , then \int\limits_0^1 \left( a - x \right)^2 f\left( x \right) dx\] equals

4a2

0

 2a2

none of these

Q 23 | Page 119

The value of \[\int\limits_{- \pi}^\pi \sin^3 x \cos^2 x\ dx\] is 

 

\[\frac{\pi^4}{2}\]
\[\frac{\pi^4}{4}\]

 0

none of these

Q 24 | Page 119
\[\int\limits_{\pi/6}^{\pi/3} \frac{1}{\sin 2x} dx\]  is equal to

 loge 3

\[\log_e \sqrt{3}\]
\[\frac{1}{2}\log\left( - 1 \right)\]

 log (−1)

 
Q 25 | Page 119
\[\int\limits_{- 1}^1 \left| 1 - x \right| dx\]  is equal to

−2

2

0

4

Q 26 | Page 119

The derivative of \[f\left( x \right) = \int\limits_{x^2}^{x^3} \frac{1}{\log_e t} dt, \left( x > 0 \right),\] is

 

\[\frac{1}{3 \ln x}\]
\[\frac{1}{3 \ln x} - \frac{1}{2 \ln x}\]

(ln x)−1 x (x − 1)

\[\frac{3 x^2}{\ln x}\]
Q 27 | Page 119

If \[I_{10} = \int\limits_0^{\pi/2} x^{10} \sin x\ dx,\]  then the value of I10 + 90I8 is

 

\[9 \left( \frac{\pi}{2} \right)^9\]
\[10 \left( \frac{\pi}{2} \right)^9\]
\[\left( \frac{\pi}{2} \right)^9\]
\[9 \left( \frac{\pi}{2} \right)^8\]
Q 28 | Page 119
\[\int\limits_0^1 \frac{x}{\left( 1 - x \right)^{54}} dx =\]
Q 29 | Page 119
\[\lim_{n \square \infty} \left\{ \frac{1}{2n + 1} + \frac{1}{2n + 2} + . . . + \frac{1}{2n + n} \right\}\] is equal to
\[\ln\left( \frac{1}{3} \right)\]
\[\ln\left( \frac{2}{3} \right)\]
\[\ln\left( \frac{3}{2} \right)\]
\[\ln\left( \frac{4}{3} \right)\]
Q 30 | Page 118

The value of the integral \[\int\limits_{- 2}^2 \left| 1 - x^2 \right| dx\] is 

 

 4

 2

−2

0

Q 31 | Page 119
\[\int\limits_0^{\pi/2} \frac{1}{1 + \cot^3 x} dx\]  is equal to

0

1

π/2

π/4

Q 32 | Page 119
\[\int\limits_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx\]  equals to
Q 33 | Page 120
\[\int\limits_0^1 \frac{d}{dx}\left\{ \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \right\} dx\] is equal to

 0

 π

π/2

π/4

Q 34 | Page 120
\[\int\limits_0^{\pi/2} x \sin x\ dx\]  is equal to

 π/4

 π/2

π

1

Q 35 | Page 120
\[\int\limits_0^{\pi/2} \sin\ 2x\ \log\ \tan x\ dx\]  is equal to 

π

 π/2

 0

Q 36 | Page 120

The value of \[\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx\] is

 

Q 37 | Page 120
\[\int\limits_0^\infty \log\left( x + \frac{1}{x} \right) \frac{1}{1 + x^2} dx =\] 

π ln 2

−π ln 2

0

\[- \frac{\pi}{2}\ln 2\]

Q 38 | Page 120

\[\int\limits_0^{2a} f\left( x \right) dx\]  is equal to

\[2 \int\limits_0^a f\left( x \right) dx\]

 0

\[\int\limits_0^a f\left( x \right) dx + \int\limits_0^a f\left( 2a - x \right) dx\]

\[\int\limits_0^a f\left( x \right) dx + \int\limits_0^{2a} f\left( 2a - x \right) dx\]
Q 39 | Page 120

If f (a + b − x) = f (x), then \[\int\limits_a^b\] x f (x) dx is equal to

\[\frac{a + b}{2} \int\limits_a^b f\left( b - x \right) dx\]

 

\[\frac{a + b}{2} \int\limits_a^b f\left( b + x \right) dx\]

 

\[\frac{b - a}{2} \int\limits_a^b f\left( x \right) dx\]
\[\frac{b + a}{2} \int\limits_a^b f\left( x \right) dx\]
Q 40 | Page 120

The value of \[\int\limits_0^1 \tan^{- 1} \left( \frac{2x - 1}{1 + x - x^2} \right) dx,\] is

1

0

−1

π/4

Q 41 | Page 120

The value of \[\int\limits_0^{\pi/2} \log\left( \frac{4 + 3 \sin x}{4 + 3 \cos x} \right) dx\] is 

 

2

\[\frac{3}{4}\]

0

−2

Q 42 | Page 120

The value of \[\int\limits_{- \pi/2}^{\pi/2} \left( x^3 + x \cos x + \tan^5 x + 1 \right) dx, \] is 

 0

2

π

1

R.D. Sharma Mathematics Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session)

R.D. Sharma solutions for Class 12 Mathematics chapter 19 - Indefinite Integrals

R.D. Sharma solutions for Class 12 Mathematics chapter 19 (Indefinite Integrals) include all questions with solution and detail explanation from Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session). 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 created 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 Indefinite Integral Problems, Definite Integrals 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 R.D. Sharma solutions for Class 12 Mathematics 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 R.D. Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer R.D. Sharma Textbook Solutions to score more in exam.

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