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# RD Sharma solutions for Class 12 Mathematics chapter 20 - Definite Integrals

## Chapter 20: Definite Integrals

Ex. 20.1Ex. 20.2Ex. 20.3Ex. 20.4Ex. 20.5Ex. 20.6Ex. Very Short AnswersEx. MCQEx. Revision Exercise

#### Chapter 20: Definite Integrals Exercise 20.1 solutions [Pages 16 - 18]

Ex. 20.1 | Q 1 | Page 16
$\int\limits_4^9 \frac{1}{\sqrt{x}} dx$
Ex. 20.1 | Q 2 | Page 16
$\int\limits_{- 2}^3 \frac{1}{x + 7} dx$
Ex. 20.1 | Q 3 | Page 16
$\int\limits_0^{1/2} \frac{1}{\sqrt{1 - x^2}} dx$
Ex. 20.1 | Q 4 | Page 16
$\int\limits_0^1 \frac{1}{1 + x^2} dx$
Ex. 20.1 | Q 5 | Page 16
$\int\limits_2^3 \frac{x}{x^2 + 1} dx$
Ex. 20.1 | Q 6 | Page 16
$\int\limits_0^\infty \frac{1}{a^2 + b^2 x^2} dx$
Ex. 20.1 | Q 7 | Page 16
$\int\limits_{- 1}^1 \frac{1}{1 + x^2} dx$
Ex. 20.1 | Q 8 | Page 16
$\int\limits_0^\infty e^{- x} dx$
Ex. 20.1 | Q 9 | Page 16
$\int\limits_0^1 \frac{x}{x + 1} dx$
Ex. 20.1 | Q 10 | Page 16
$\int\limits_0^{\pi/2} \left( \sin x + \cos x \right) dx$
Ex. 20.1 | Q 11 | Page 16

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

Ex. 20.1 | Q 12 | Page 16
$\int\limits_0^{\pi/4} \sec x dx$
Ex. 20.1 | Q 13 | Page 16
$\int\limits_{\pi/6}^{\pi/4} cosec\ x\ dx$
Ex. 20.1 | Q 14 | Page 16
$\int\limits_0^1 \frac{1 - x}{1 + x} dx$
Ex. 20.1 | Q 15 | Page 16
$\int\limits_0^\pi \frac{1}{1 + \sin x} dx$
Ex. 20.1 | Q 16 | Page 16
$\int\limits_{- \pi/4}^{\pi/4} \frac{1}{1 + \sin x} dx$
Ex. 20.1 | Q 17 | Page 16
$\int\limits_0^{\pi/2} \cos^2 x\ dx$
Ex. 20.1 | Q 18 | Page 16
$\int\limits_0^{\pi/2} \cos^3 x\ dx$
Ex. 20.1 | Q 19 | Page 16
$\int\limits_0^{\pi/6} \cos x \cos 2x\ dx$
Ex. 20.1 | Q 20 | Page 16
$\int\limits_0^{\pi/2} \sin x \sin 2x\ dx$
Ex. 20.1 | Q 21 | Page 16
$\int\limits_{\pi/3}^{\pi/4} \left( \tan x + \cot x \right)^2 dx$
Ex. 20.1 | Q 22 | Page 16
$\int\limits_0^{\pi/2} \cos^4\ x\ dx$

Ex. 20.1 | Q 23 | Page 16
$\int\limits_0^{\pi/2} \left( a^2 \cos^2 x + b^2 \sin^2 x \right) dx$
Ex. 20.1 | Q 24 | Page 16
$\int\limits_0^{\pi/2} \sqrt{1 + \sin x}\ dx$
Ex. 20.1 | Q 25 | Page 16
$\int\limits_0^{\pi/2} \sqrt{1 + \cos x}\ dx$
Ex. 20.1 | Q 26 | Page 16

Evaluate the following definite integrals:

$\int_0^\frac{\pi}{2} x^2 \sin\ x\ dx$
Ex. 20.1 | Q 27 | Page 17
$\int\limits_0^{\pi/2} x \cos\ x\ dx$
Ex. 20.1 | Q 28 | Page 17
$\int\limits_0^{\pi/2} x^2 \cos\ x\ dx$
Ex. 20.1 | Q 29 | Page 17
$\int\limits_0^{\pi/4} x^2 \sin\ x\ dx$
Ex. 20.1 | Q 30 | Page 17
$\int\limits_0^{\pi/2} x^2 \cos\ 2x\ dx$
Ex. 20.1 | Q 31 | Page 17
$\int\limits_0^{\pi/2} x^2 \cos^2 x\ dx$
Ex. 20.1 | Q 32 | Page 17
$\int\limits_1^2 \log\ x\ dx$
Ex. 20.1 | Q 33 | Page 17
$\int\limits_1^3 \frac{\log x}{\left( x + 1 \right)^2} dx$
Ex. 20.1 | Q 34 | Page 17
$\int\limits_1^e \frac{e^x}{x} \left( 1 + x \log x \right) dx$
Ex. 20.1 | Q 35 | Page 17
$\int\limits_1^e \frac{\log x}{x} dx$
Ex. 20.1 | Q 36 | Page 17
$\int\limits_e^{e^2} \left\{ \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right\} dx$
Ex. 20.1 | Q 37 | Page 17
$\int\limits_1^2 \frac{x + 3}{x \left( x + 2 \right)} dx$
Ex. 20.1 | Q 38 | Page 17
$\int\limits_0^1 \frac{2x + 3}{5 x^2 + 1} dx$
Ex. 20.1 | Q 39 | Page 17
$\int\limits_0^2 \frac{1}{4 + x - x^2} dx$
Ex. 20.1 | Q 40 | Page 17
$\int\limits_0^1 \frac{1}{2 x^2 + x + 1} dx$
Ex. 20.1 | Q 41 | Page 17
$\int\limits_0^1 \sqrt{x \left( 1 - x \right)} dx$
Ex. 20.1 | Q 42 | Page 17
$\int\limits_0^2 \frac{1}{\sqrt{3 + 2x - x^2}} dx$
Ex. 20.1 | Q 43 | Page 17
$\int\limits_0^4 \frac{1}{\sqrt{4x - x^2}} dx$
Ex. 20.1 | Q 44 | Page 17
$\int\limits_{- 1}^1 \frac{1}{x^2 + 2x + 5} dx$
Ex. 20.1 | Q 45 | Page 17
$\int\limits_1^4 \frac{x^2 + x}{\sqrt{2x + 1}} dx$
Ex. 20.1 | Q 46 | Page 17
$\int\limits_0^1 x \left( 1 - x \right)^5 dx$
Ex. 20.1 | Q 47 | Page 17
$\int\limits_1^2 \left( \frac{x - 1}{x^2} \right) e^x dx$
Ex. 20.1 | Q 48 | Page 17
$\int\limits_0^1 \left( x e^{2x} + \sin\frac{\ pix}{2} \right) dx$
Ex. 20.1 | Q 49 | Page 17

$\int\limits_0^1 \left( x e^x + \cos\frac{\pi x}{4} \right) dx$

Ex. 20.1 | Q 50 | Page 17
$\int\limits_{\pi/2}^\pi e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx$
Ex. 20.1 | Q 51 | Page 17
$\int\limits_0^{2\pi} e^{x/2} \sin\left( \frac{x}{2} + \frac{\pi}{4} \right) dx$
Ex. 20.1 | Q 52 | Page 17
$\int\limits_0^{2\pi} e^x \cos\left( \frac{\pi}{4} + \frac{x}{2} \right) dx$
Ex. 20.1 | Q 53 | Page 17
$\int_0^\pi e^{2x} \cdot \sin\left( \frac{\pi}{4} + x \right) dx$
Ex. 20.1 | Q 54 | Page 17
$\int\limits_0^1 \frac{1}{\sqrt{1 + x} - \sqrt{x}} dx$
Ex. 20.1 | Q 55 | Page 17
$\int\limits_1^2 \frac{x}{\left( x + 1 \right) \left( x + 2 \right)} dx$
Ex. 20.1 | Q 56 | Page 17
$\int\limits_0^{\pi/2} \sin^3 x\ dx$
Ex. 20.1 | Q 57 | Page 17
$\int\limits_0^\pi \left( \sin^2 \frac{x}{2} - \cos^2 \frac{x}{2} \right) dx$
Ex. 20.1 | Q 58 | Page 17
$\int\limits_1^2 e^{2x} \left( \frac{1}{x} - \frac{1}{2 x^2} \right) dx$
Ex. 20.1 | Q 59 | Page 17

Evaluate the following definite integral:

$\int_0^1 \frac{1}{\sqrt{\left( x - 1 \right)\left( 2 - x \right)}}dx$
Ex. 20.1 | Q 60 | Page 17

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

Ex. 20.1 | Q 61 | Page 18

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

Ex. 20.1 | Q 62 | Page 18
$\int_\pi^\frac{3\pi}{2} \sqrt{1 - \cos2x}dx$
Ex. 20.1 | Q 63 | Page 18
$\int_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx$
Ex. 20.1 | Q 64 | Page 18
$\int_0^\frac{\pi}{4} \left( \tan x + \cot x \right)^{- 2} dx$
Ex. 20.1 | Q 65 | Page 18
$\int_0^1 x\log\left( 1 + 2x \right)dx$
Ex. 20.1 | Q 66 | Page 18
$\int_\frac{\pi}{6}^\frac{\pi}{3} \left( \tan x + \cot x \right)^2 dx$
Ex. 20.1 | Q 67 | Page 18
$\int_0^\frac{\pi}{4} \left( a^2 \cos^2 x + b^2 \sin^2 x \right)dx$
Ex. 20.1 | Q 68 | Page 18
$\int_0^1 \frac{1}{1 + 2x + 2 x^2 + 2 x^3 + x^4}dx$

#### Chapter 20: Definite Integrals Exercise 20.2 solutions [Pages 38 - 40]

Ex. 20.2 | Q 1 | Page 38
$\int\limits_2^4 \frac{x}{x^2 + 1} dx$
Ex. 20.2 | Q 2 | Page 38
$\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx$
Ex. 20.2 | Q 3 | Page 38
$\int\limits_1^2 \frac{3x}{9 x^2 - 1} dx$
Ex. 20.2 | Q 4 | Page 38
$\int\limits_0^{\pi/2} \frac{1}{5 \cos x + 3 \sin x} dx$
Ex. 20.2 | Q 5 | Page 38
$\int\limits_0^a \frac{x}{\sqrt{a^2 + x^2}} dx$
Ex. 20.2 | Q 6 | Page 38
$\int\limits_0^1 \frac{e^x}{1 + e^{2x}} dx$
Ex. 20.2 | Q 7 | Page 38
$\int\limits_0^1 x e^{x^2} dx$
Ex. 20.2 | Q 8 | Page 38
$\int\limits_1^3 \frac{\cos \left( \log x \right)}{x} dx$
Ex. 20.2 | Q 9 | Page 38
$\int\limits_0^1 \frac{2x}{1 + x^4} dx$
Ex. 20.2 | Q 10 | Page 38
$\int\limits_0^a \sqrt{a^2 - x^2} dx$
Ex. 20.2 | Q 11 | Page 39
$\int\limits_0^{\pi/2} \sqrt{\sin \phi} \cos^5 \phi\ d\phi$

Ex. 20.2 | Q 12 | Page 39
$\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx$
Ex. 20.2 | Q 13 | Page 39
$\int\limits_0^{\pi/2} \frac{\sin \theta}{\sqrt{1 + \cos \theta}} d\theta$
Ex. 20.2 | Q 14 | Page 39
$\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx$
Ex. 20.2 | Q 15 | Page 39
$\int\limits_0^1 \frac{\sqrt{\tan^{- 1} x}}{1 + x^2} dx$
Ex. 20.2 | Q 16 | Page 39
$\int\limits_0^2 x\sqrt{x + 2}\ dx$
Ex. 20.2 | Q 17 | Page 39
$\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx$
Ex. 20.2 | Q 18 | Page 39
$\int\limits_0^{\pi/2} \frac{\sin x \cos x}{1 + \sin^4 x} dx$
Ex. 20.2 | Q 19 | Page 39
$\int\limits_0^{\pi/2} \frac{dx}{a \cos x + b \sin x}a, b > 0$
Ex. 20.2 | Q 20 | Page 39
$\int\limits_0^{\pi/2} \frac{1}{5 + 4 \sin x} dx$
Ex. 20.2 | Q 21 | Page 39
$\int\limits_0^\pi \frac{\sin x}{\sin x + \cos x} dx$
Ex. 20.2 | Q 22 | Page 39
$\int\limits_0^\pi \frac{1}{3 + 2 \sin x + \cos x} dx$
Ex. 20.2 | Q 23 | Page 39
$\int\limits_0^1 \tan^{- 1} x\ dx$
Ex. 20.2 | Q 24 | Page 39
$\int_0^\frac{1}{2} \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}dx$
Ex. 20.2 | Q 25 | Page 39
$\int\limits_0^{\pi/4} \left( \sqrt{\tan}x + \sqrt{\cot}x \right) dx$
Ex. 20.2 | Q 26 | Page 39
$\int\limits_0^{\pi/4} \frac{\tan^3 x}{1 + \cos 2x} dx$
Ex. 20.2 | Q 27 | Page 39
$\int\limits_0^\pi \frac{1}{5 + 3 \cos x} dx$
Ex. 20.2 | Q 28 | Page 39
$\int\limits_0^{\pi/2} \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx$
Ex. 20.2 | Q 29 | Page 39
$\int\limits_0^{\pi/2} \frac{x + \sin x}{1 + \cos x} dx$
Ex. 20.2 | Q 30 | Page 39
$\int\limits_0^1 \frac{\tan^{- 1} x}{1 + x^2} dx$
Ex. 20.2 | Q 31 | Page 39
$\int_0^\frac{\pi}{4} \frac{\sin x + \cos x}{3 + \sin2x}dx$
Ex. 20.2 | Q 32 | Page 39
$\int\limits_0^1 x \tan^{- 1} x\ dx$
Ex. 20.2 | Q 33 | Page 39
$\int\limits_0^1 \frac{1 - x^2}{x^4 + x^2 + 1} dx$
Ex. 20.2 | Q 34 | Page 39
$\int\limits_0^1 \frac{24 x^3}{\left( 1 + x^2 \right)^4} dx$
Ex. 20.2 | Q 35 | Page 39
$\int\limits_4^{12} x \left( x - 4 \right)^{1/3} dx$
Ex. 20.2 | Q 36 | Page 39
$\int\limits_0^{\pi/2} x^2 \sin\ x\ dx$
Ex. 20.2 | Q 37 | Page 39
$\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx$
Ex. 20.2 | Q 38 | Page 39
$\int\limits_0^1 \frac{1 - x^2}{\left( 1 + x^2 \right)^2} dx$
Ex. 20.2 | Q 39 | Page 39
$\int\limits_{- 1}^1 5 x^4 \sqrt{x^5 + 1} dx$
Ex. 20.2 | Q 40 | Page 39
$\int_0^\frac{\pi}{2} \frac{\cos^2 x}{1 + 3 \sin^2 x}dx$
Ex. 20.2 | Q 41 | Page 39
$\int\limits_0^{\pi/4} \sin^3 2t \cos 2t\ dt$
Ex. 20.2 | Q 42 | Page 39
$\int\limits_0^\pi 5 \left( 5 - 4 \cos \theta \right)^{1/4} \sin \theta\ d \theta$
Ex. 20.2 | Q 43 | Page 39
$\int\limits_0^{\pi/6} \cos^{- 3} 2 \theta \sin 2\ \theta\ d\ \theta$
Ex. 20.2 | Q 44 | Page 39

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

Ex. 20.2 | Q 45 | Page 40
$\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx$
Ex. 20.2 | Q 46 | Page 40
$\int\limits_0^{\pi/2} \cos^5 x\ dx$
Ex. 20.2 | Q 47 | Page 40
$\int\limits_4^9 \frac{\sqrt{x}}{\left( 30 - x^{3/2} \right)^2} dx$
Ex. 20.2 | Q 48 | Page 40
$\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx$
Ex. 20.2 | Q 49 | Page 40
$\int\limits_0^{\pi/2} 2 \sin x \cos x \tan^{- 1} \left( \sin x \right) dx$
Ex. 20.2 | Q 50 | Page 39
$\int\limits_0^{\pi/2} \sin 2x \tan^{- 1} \left( \sin x \right) dx$
Ex. 20.2 | Q 51 | Page 40
$\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx$
Ex. 20.2 | Q 52 | Page 40
$\int\limits_0^a \sin^{- 1} \sqrt{\frac{x}{a + x}} dx$
Ex. 20.2 | Q 53 | Page 40
$\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{3/2}} dx$
Ex. 20.2 | Q 54 | Page 40
$\int\limits_0^a x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} dx$
Ex. 20.2 | Q 55 | Page 40
$\int\limits_{- a}^a \sqrt{\frac{a - x}{a + x}} dx$
Ex. 20.2 | Q 56 | Page 40
$\int\limits_0^{\pi/2} \frac{\sin x \cos x}{\cos^2 x + 3 \cos x + 2} dx$
Ex. 20.2 | Q 57 | Page 40
$\int_0^\frac{\pi}{2} \frac{\tan x}{1 + m^2 \tan^2 x}dx$
Ex. 20.2 | Q 58 | Page 40
$\int_0^\frac{1}{2} \frac{1}{\left( 1 + x^2 \right)\sqrt{1 - x^2}}dx$
Ex. 20.2 | Q 59 | Page 40
$\int_\frac{1}{3}^1 \frac{\left( x - x^3 \right)^\frac{1}{3}}{x^4}dx$
Ex. 20.2 | Q 60 | Page 40
$\int_0^\frac{\pi}{4} \frac{\sin^2 x \cos^2 x}{\left( \sin^3 x + \cos^3 x \right)^2}dx$
Ex. 20.2 | Q 61 | Page 40
$\int_0^\frac{\pi}{2} \sqrt{\cos x - \cos^3 x}\left( \sec^2 x - 1 \right) \cos^2 xdx$
Ex. 20.2 | Q 62 | Page 40
$\int_0^\frac{\pi}{2} \frac{\cos x}{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^n}dx$

#### Chapter 20: Definite Integrals Exercise 20.3 solutions [Pages 55 - 56]

Ex. 20.3 | Q 1.1 | Page 55
$\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}4x + 3 & , & \text{if }1 \leq x \leq 2 \\3x + 5 & , & \text{if }2 \leq x \leq 4\end{cases}$

Ex. 20.3 | Q 1.2 | Page 55
$\int\limits_0^9 f\left( x \right) dx, where f\left( x \right) \begin{cases}\sin x & , & 0 \leq x \leq \pi/2 \\ 1 & , & \pi/2 \leq x \leq 3 \\ e^{x - 3} & , & 3 \leq x \leq 9\end{cases}$
Ex. 20.3 | Q 1.3 | Page 55

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

Ex. 20.3 | Q 2 | Page 56

Evaluate the following integral:

$\int\limits_{- 4}^4 \left| x + 2 \right| dx$
Ex. 20.3 | Q 3 | Page 56

Evaluate the following integral:

$\int\limits_{- 3}^3 \left| x + 1 \right| dx$
Ex. 20.3 | Q 4 | Page 56

Evaluate the following integral:

$\int\limits_{- 1}^1 \left| 2x + 1 \right| dx$
Ex. 20.3 | Q 5 | Page 56

Evaluate the following integral:

$\int\limits_{- 2}^2 \left| 2x + 3 \right| dx$
Ex. 20.3 | Q 6 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 7 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 8 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 9 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 10 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 11 | Page 56

Evaluate the following integral:

$\int\limits_0^{\pi/2} \left| \cos 2x \right| dx$
Ex. 20.3 | Q 12 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 13 | Page 56

Evaluate the following integral:

$\int\limits_{- \pi/4}^{\pi/4} \left| \sin x \right| dx$
Ex. 20.3 | Q 14 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 15 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 16 | Page 56

Evaluate the following integral:

$\int\limits_0^4 \left| x - 1 \right| dx$
Ex. 20.3 | Q 17 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 18 | Page 56

Evaluate the following integral:

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

Ex. 20.3 | Q 19 | Page 56

Evaluate the following integral:

$\int\limits_0^4 \left( \left| x \right| + \left| x - 2 \right| + \left| x - 4 \right| \right) dx$
Ex. 20.3 | Q 20 | Page 56
$\int_{- 1}^2 \left( \left| x + 1 \right| + \left| x \right| + \left| x - 1 \right| \right)dx$

Ex. 20.3 | Q 21 | Page 56
$\int_{- 2}^2 x e^\left| x \right| dx$
Ex. 20.3 | Q 22 | Page 56
$\int_{- \frac{\pi}{4}}^\frac{\pi}{2} \sin x\left| \sin x \right|dx$

Ex. 20.3 | Q 23 | Page 56
$\int_0^\pi \cos x\left| \cos x \right|dx$
Ex. 20.3 | Q 24 | Page 56
$\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left( 2\sin\left| x \right| + \cos\left| x \right| \right)dx$
Ex. 20.3 | Q 25 | Page 56
$\int_{- \frac{\pi}{2}}^\pi \sin^{- 1} \left( \sin x \right)dx$
Ex. 20.3 | Q 26 | Page 56
$\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{- \frac{\pi}{2}}{\sqrt{\cos x \sin^2 x}}dx$
Ex. 20.3 | Q 27 | Page 56
$\int_0^2 2x\left[ x \right]dx$
Ex. 20.3 | Q 28 | Page 56
$\int_0^{2\pi} \cos^{- 1} \left( \cos x \right)dx$

#### Chapter 20: Definite Integrals Exercise 20.4 solutions [Page 61]

Ex. 20.4 | Q 1 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 2 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 3 | Page 61

Evaluate each of the following integral:

$\int_\frac{\pi}{6}^\frac{\pi}{3} \frac{\sqrt{\tan x}}{\sqrt{\tan x} + \sqrt{\cot x}}dx$
Ex. 20.4 | Q 4 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 5 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 6 | Page 61

Evaluate each of the following integral:

$\int_{- a}^a \frac{1}{1 + a^x}dx$, a > 0
Ex. 20.4 | Q 7 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 8 | Page 61

Evaluate each of the following integral:

$\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{\cos^2 x}{1 + e^x}dx$
Ex. 20.4 | Q 9 | Page 61

Evaluate each of the following integral:

$\int_{- \frac{\pi}{4}}^\frac{\pi}{4} \frac{x^{11} - 3 x^9 + 5 x^7 - x^5 + 1}{\cos^2 x}dx$
Ex. 20.4 | Q 10 | Page 61

Evaluate each of the following integral:

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

Ex. 20.4 | Q 11 | Page 61
$\int\limits_0^{\pi/2} \left( 2 \log \cos x - \log \sin 2x \right) dx$

Ex. 20.4 | Q 12 | Page 61
$\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx$
Ex. 20.4 | Q 13 | Page 61
$\int\limits_0^5 \frac{\sqrt[4]{x + 4}}{\sqrt[4]{x + 4} + \sqrt[4]{9 - x}} dx$
Ex. 20.4 | Q 14 | Page 61
$\int\limits_0^7 \frac{\sqrt[3]{x}}{\sqrt[3]{x} + \sqrt[3]{7} - x} dx$
Ex. 20.4 | Q 15 | Page 61
$\int\limits_{\pi/6}^{\pi/3} \frac{1}{1 + \sqrt{\tan x}} dx$
Ex. 20.4 | Q 16 | Page 61

If  $f\left( a + b - x \right) = f\left( x \right)$ , then prove that $\int_a^b xf\left( x \right)dx = \frac{a + b}{2} \int_a^b f\left( x \right)dx$

#### Chapter 20: Definite Integrals Exercise 20.5 solutions [Pages 94 - 96]

Ex. 20.5 | Q 1 | Page 94
$\int\limits_0^{\pi/2} \frac{1}{1 + \tan x}$

Ex. 20.5 | Q 2 | Page 94
$\int\limits_0^{\pi/2} \frac{1}{1 + \cot x} dx$
Ex. 20.5 | Q 3 | Page 94
$\int\limits_0^{\pi/2} \frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}} dx$
Ex. 20.5 | Q 4 | Page 94
$\int\limits_0^{\pi/2} \frac{\sin^{3/2} x}{\sin^{3/2} x + \cos^{3/2} x} dx$
Ex. 20.5 | Q 5 | Page 94
$\int\limits_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x} dx$

Ex. 20.5 | Q 6 | Page 94
$\int\limits_0^{\pi/2} \frac{1}{1 + \sqrt{\tan x}} dx$
Ex. 20.5 | Q 7 | Page 95
$\int\limits_0^a \frac{1}{x + \sqrt{a^2 - x^2}} dx$
Ex. 20.5 | Q 8 | Page 95
$\int\limits_0^\infty \frac{\log x}{1 + x^2} dx$
Ex. 20.5 | Q 9 | Page 95
$\int\limits_0^1 \frac{\log\left( 1 + x \right)}{1 + x^2} dx$

Ex. 20.5 | Q 10 | Page 95
$\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx$
Ex. 20.5 | Q 11 | Page 95
$\int\limits_0^\pi \frac{x \tan x}{\sec x \ cosec x} dx$
Ex. 20.5 | Q 12 | Page 95
$\int\limits_0^\pi x \sin x \cos^4 x\ dx$
Ex. 20.5 | Q 13 | Page 95
$\int\limits_0^\pi x \sin^3 x\ dx$
Ex. 20.5 | Q 14 | Page 95
$\int\limits_0^\pi x \log \sin x\ dx$
Ex. 20.5 | Q 15 | Page 95
$\int\limits_0^\pi \frac{x \sin x}{1 + \sin x} dx$
Ex. 20.5 | Q 16 | Page 95
$\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx, 0 < \alpha < \pi$
Ex. 20.5 | Q 17 | Page 95
$\int\limits_0^\pi x \cos^2 x\ dx$
Ex. 20.5 | Q 18 | Page 95

Evaluate the following integral:

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

Ex. 20.5 | Q 19 | Page 95

Evaluate the following integral:

$\int_0^\frac{\pi}{2} \frac{\tan^7 x}{\tan^7 x + \cot^7 x}dx$
Ex. 20.5 | Q 20 | Page 95

Evaluate the following integral:

$\int_2^8 \frac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}}dx$
Ex. 20.5 | Q 21 | Page 95

Evaluate the following integral:

$\int_0^\pi x\sin x \cos^2 xdx$
Ex. 20.5 | Q 22 | Page 95
$\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx$
Ex. 20.5 | Q 23 | Page 95
$\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx$
Ex. 20.5 | Q 24 | Page 95
$\int\limits_{- \pi/2}^{\pi/2} \sin^4 x\ dx$
Ex. 20.5 | Q 25 | Page 95
$\int\limits_{- 1}^1 \log\left( \frac{2 - x}{2 + x} \right) dx$
Ex. 20.5 | Q 26 | Page 95
$\int\limits_{- \pi/4}^{\pi/4} \sin^2 x\ dx$
Ex. 20.5 | Q 27 | Page 95
$\int\limits_0^\pi \log\left( 1 - \cos x \right) dx$
Ex. 20.5 | Q 28 | Page 95
$\int\limits_{- \pi/2}^{\pi/2} \log\left( \frac{2 - \sin x}{2 + \sin x} \right) dx$
Ex. 20.5 | Q 29 | Page 95

Evaluate the following integral:

$\int_{- \pi}^\pi \frac{2x\left( 1 + \sin x \right)}{1 + \cos^2 x}dx$
Ex. 20.5 | Q 30 | Page 95

Evaluate the following integral:

$\int_{- a}^a \log\left( \frac{a - \sin\theta}{a + \sin\theta} \right)d\theta$
Ex. 20.5 | Q 31 | Page 95

Evaluate the following integral:

$\int_{- 2}^2 \frac{3 x^3 + 2\left| x \right| + 1}{x^2 + \left| x \right| + 1}dx$
Ex. 20.5 | Q 32 | Page 95

Evaluate the following integral:

$\int_{- \frac{3\pi}{2}}^{- \frac{\pi}{2}} \left\{ \sin^2 \left( 3\pi + x \right) + \left( \pi + x \right)^3 \right\}dx$
Ex. 20.5 | Q 33 | Page 95
$\int\limits_0^2 x\sqrt{2 - x} dx$
Ex. 20.5 | Q 34 | Page 95
$\int\limits_0^1 \log\left( \frac{1}{x} - 1 \right) dx$

Ex. 20.5 | Q 35 | Page 95

Evaluate the following integral:

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

Ex. 20.5 | Q 36 | Page 95

Evaluate the following integral:

$\int_0^\pi \left( \frac{x}{1 + \sin^2 x} + \cos^7 x \right)dx$
Ex. 20.5 | Q 37 | Page 95

Evaluate

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

Ex. 20.5 | Q 38 | Page 95

Evaluate the following integral:

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

Ex. 20.5 | Q 39 | Page 95

Evaluate the following integral:

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

Ex. 20.5 | Q 40 | Page 95

int_0^(2a)f(x)dx

Ex. 20.5 | Q 41 | Page 95
$\int_0^1 | x\sin \pi x | dx$
Ex. 20.5 | Q 42 | Page 95

Evaluate :

$\int\limits_0^{3/2} \left| x \sin \pi x \right|dx$
Ex. 20.5 | Q 43 | Page 96

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

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

Ex. 20.5 | Q 44 | Page 96

If f(2a − x) = −f(x), prove that

$\int\limits_0^{2a} f\left( x \right) dx = 0 .$
Ex. 20.5 | Q 45.1 | Page 96

If f is an integrable function, show that

$\int\limits_{- a}^a f\left( x^2 \right) dx = 2 \int\limits_0^a f\left( x^2 \right) dx$

Ex. 20.5 | Q 45.2 | Page 96

If f is an integrable function, show that

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

Ex. 20.5 | Q 46 | Page 96

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

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

Ex. 20.5 | Q 47 | Page 96

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

$\int_a^b xf\left( x \right)dx = \left( \frac{a + b}{2} \right) \int_a^b f\left( x \right)dx$
Ex. 20.5 | Q 48 | Page 96

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

$\int\limits_{- a}^a f\left( x \right) dx = \int\limits_0^a \left\{ f\left( x \right) + f\left( - x \right) \right\} dx$
Ex. 20.5 | Q 49 | Page 96

Prove that:

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

#### Chapter 20: Definite Integrals Exercise 20.6 solutions [Pages 110 - 111]

Ex. 20.6 | Q 1 | Page 110
$\int\limits_0^3 \left( x + 4 \right) dx$
Ex. 20.6 | Q 2 | Page 110
$\int\limits_0^2 \left( x + 3 \right) dx$
Ex. 20.6 | Q 3 | Page 110
$\int\limits_1^3 \left( 3x - 2 \right) dx$
Ex. 20.6 | Q 4 | Page 110
$\int\limits_{- 1}^1 \left( x + 3 \right) dx$
Ex. 20.6 | Q 5 | Page 110
$\int\limits_0^5 \left( x + 1 \right) dx$
Ex. 20.6 | Q 6 | Page 110
$\int\limits_1^3 \left( 2x + 3 \right) dx$
Ex. 20.6 | Q 7 | Page 110
$\int\limits_3^5 \left( 2 - x \right) dx$
Ex. 20.6 | Q 8 | Page 110
$\int\limits_0^2 \left( x^2 + 1 \right) dx$
Ex. 20.6 | Q 9 | Page 110
$\int\limits_1^2 x^2 dx$
Ex. 20.6 | Q 10 | Page 110
$\int\limits_2^3 \left( 2 x^2 + 1 \right) dx$
Ex. 20.6 | Q 11 | Page 110
$\int\limits_1^2 \left( x^2 - 1 \right) dx$
Ex. 20.6 | Q 12 | Page 110
$\int\limits_0^2 \left( x^2 + 4 \right) dx$
Ex. 20.6 | Q 13 | Page 111
$\int\limits_1^4 \left( x^2 - x \right) dx$
Ex. 20.6 | Q 14 | Page 111
$\int\limits_0^1 \left( 3 x^2 + 5x \right) dx$
Ex. 20.6 | Q 15 | Page 111
$\int\limits_0^2 e^x dx$
Ex. 20.6 | Q 16 | Page 111
$\int\limits_a^b e^x dx$
Ex. 20.6 | Q 17 | Page 111
$\int\limits_a^b \cos\ x\ dx$
Ex. 20.6 | Q 18 | Page 111
$\int\limits_0^{\pi/2} \sin x\ dx$
Ex. 20.6 | Q 19 | Page 111
$\int\limits_0^{\pi/2} \cos x\ dx$
Ex. 20.6 | Q 20 | Page 111
$\int\limits_1^4 \left( 3 x^2 + 2x \right) dx$
Ex. 20.6 | Q 21 | Page 111
$\int\limits_0^2 \left( 3 x^2 - 2 \right) dx$
Ex. 20.6 | Q 22 | Page 111
$\int\limits_0^2 \left( x^2 + 2 \right) dx$
Ex. 20.6 | Q 23 | Page 111
$\int\limits_0^4 \left( x + e^{2x} \right) dx$
Ex. 20.6 | Q 23 | Page 111
$\int\limits_0^2 \left( x^2 + 2x + 1 \right) dx$
Ex. 20.6 | Q 24 | Page 111
$\int\limits_0^2 \left( x^2 + x \right) dx$
Ex. 20.6 | Q 25 | Page 111
$\int\limits_0^2 \left( x^2 + 2x + 1 \right) dx$
Ex. 20.6 | Q 26 | Page 111
$\int\limits_0^3 \left( 2 x^2 + 3x + 5 \right) dx$
Ex. 20.6 | Q 27 | Page 111
$\int\limits_a^b x\ dx$
Ex. 20.6 | Q 28 | Page 111
$\int\limits_0^5 \left( x + 1 \right) dx$
Ex. 20.6 | Q 29 | Page 111
$\int\limits_2^3 x^2 dx$
Ex. 20.6 | Q 30 | Page 111
$\int\limits_1^4 \left( x^2 - x \right) dx$
Ex. 20.6 | Q 31 | Page 111
$\int\limits_0^2 \left( x^2 - x \right) dx$
Ex. 20.6 | Q 32 | Page 111
$\int\limits_1^3 \left( 2 x^2 + 5x \right) dx$
Ex. 20.6 | Q 33 | Page 111

Evaluate the following integrals as limit of sums:

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

#### Chapter 20: Definite Integrals Exercise Very Short Answers solutions [Pages 111 - 116]

Ex. Very Short Answers | Q 1 | Page 115
$\int\limits_0^{\pi/2} \sin^2 x\ dx .$
Ex. Very Short Answers | Q 2 | Page 115
$\int\limits_0^{\pi/2} \cos^2 x\ dx .$
Ex. Very Short Answers | Q 3 | Page 115
$\int\limits_{- \pi/2}^{\pi/2} \sin^2 x\ dx .$
Ex. Very Short Answers | Q 4 | Page 115
$\int\limits_{- \pi/2}^{\pi/2} \cos^2 x\ dx .$
Ex. Very Short Answers | Q 5 | Page 111
$\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx .$
Ex. Very Short Answers | Q 6 | Page 115
$\int\limits_{- \pi/2}^{\pi/2} x \cos^2 x\ dx .$

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

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

Evaluate each of the following integral:

$\int_0^\frac{\pi}{4} \tan\ xdx$

Ex. Very Short Answers | Q 23 | Page 115
$\int\limits_2^3 \frac{1}{x}dx$
Ex. Very Short Answers | Q 24 | Page 115
$\int\limits_0^2 \sqrt{4 - x^2} dx$
Ex. Very Short Answers | Q 25 | Page 115
$\int\limits_0^1 \frac{2x}{1 + x^2} dx$
Ex. Very Short Answers | Q 26 | Page 115

Evaluate each of the following  integral:

$\int_0^1 x e^{x^2} dx$

Ex. Very Short Answers | Q 27 | Page 115

Evaluate each of the following integral:

$\int_0^\frac{\pi}{4} \sin2xdx$
Ex. Very Short Answers | Q 28 | Page 115

Evaluate each of the following integral:

$\int_e^{e^2} \frac{1}{x\log x}dx$
Ex. Very Short Answers | Q 29 | Page 115

Evaluate each of the following integral:

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

Ex. Very Short Answers | Q 30 | Page 115

Solve each of the following integral:

$\int_2^4 \frac{x}{x^2 + 1}dx$
Ex. Very Short Answers | Q 31 | Page 116

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

Ex. Very Short Answers | Q 32 | Page 116

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

Ex. Very Short Answers | Q 33 | Page 116

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

Ex. Very Short Answers | Q 34 | Page 116

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

Ex. Very Short Answers | 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.
Ex. Very Short Answers | Q 36 | Page 116

Evaluate :

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

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

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

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

#### Chapter 20: Definite Integrals Exercise MCQ solutions [Pages 117 - 120]

Ex. MCQ | Q 1 | Page 117
$\int\limits_0^1 \sqrt{x \left( 1 - x \right)} dx$ equals
• π/2

• π/4

• π/6

• π/8

Ex. MCQ | Q 2 | Page 117

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

• 0

• 1/2

• 2

• 3/2

Ex. MCQ | Q 3 | Page 117

The value of $\int\limits_0^\pi \frac{x \tan x}{\sec x + \cos x} dx$ is __________ .

• $\frac{\pi^2}{4}$
• $\frac{\pi^2}{2}$
• $\frac{3 \pi^2}{2}$
• $\frac{\pi^2}{3}$

Ex. MCQ | Q 4 | Page 117

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

• 0

• 2

• 8

• 4

Ex. MCQ | 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

Ex. MCQ | 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

Ex. MCQ | Q 7 | Page 117

$\int_0^\frac{\pi^2}{4} \frac{\sin\sqrt{x}}{\sqrt{x}} dx$ equals

• 2

• 1

• π/4

• π2/8

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

$\int\limits_0^{\pi/2} \frac{1}{2 + \cos x} dx$ equals

• $\frac{1}{3} \tan^{- 1} \left( \frac{1}{\sqrt{3}} \right)$
• $\frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{1}{\sqrt{3}} \right)$
• $\sqrt{3} \tan^{- 1} \left( \sqrt{3} \right)$

• $2\sqrt{3} \tan^{- 1} \sqrt{3}$
Ex. MCQ | 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 these

Ex. MCQ | 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

Ex. MCQ | 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

Ex. MCQ | 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 $\int\limits_0^\infty \frac{dx}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)},$

• $\frac{\pi}{60}$
• $\frac{\pi}{20}$
• $\frac{\pi}{40}$
• $\frac{\pi}{80}$
Ex. MCQ | Q 14 | Page 118
$\int\limits_1^e \log x\ dx =$
• 1

•  e − 1

• e + 1

•  0

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

Ex. MCQ | Q 17 | Page 118

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

• $\frac{\pi}{2}$
• $\frac{\pi}{4}$
• $\frac{\pi}{6}$
• $\frac{\pi}{3}$
Ex. MCQ | Q 18 | Page 118
$\int\limits_{- \pi/2}^{\pi/2} \sin\left| x \right| dx$  is equal to
•  1

• 2

• − 1

• − 2

Ex. MCQ | 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

Ex. MCQ | Q 20 | Page 118

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

•  1

• e − 1

• 0

• − 1

Ex. MCQ | 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

Ex. MCQ | 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

Ex. MCQ | 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

Ex. MCQ | 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)

Ex. MCQ | Q 25 | Page 119
$\int\limits_{- 1}^1 \left| 1 - x \right| dx$  is equal to
• −2

• 2

• 0

• 4

Ex. MCQ | 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)

• $\frac{3 x^2}{\ln x}$
Ex. MCQ | Q 27 | Page 119

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

• $9 \left( \frac{\pi}{2} \right)^9$
• $10 \left( \frac{\pi}{2} \right)^9$
• $\left( \frac{\pi}{2} \right)^9$
• $9 \left( \frac{\pi}{2} \right)^8$
Ex. MCQ | Q 28 | Page 119
$\int\limits_0^1 \frac{x}{\left( 1 - x \right)^\frac{5}{4}} dx =$
• 15/16

• 3/16

• -3/16

• -16/3

Ex. MCQ | Q 29 | Page 119
$\lim_{n \to \infty} \left\{ \frac{1}{2n + 1} + \frac{1}{2n + 2} + . . . + \frac{1}{2n + n} \right\}$ is equal to
• $\ln\left( \frac{1}{3} \right)$
• $\ln\left( \frac{2}{3} \right)$
• $\ln\left( \frac{3}{2} \right)$
• $\ln\left( \frac{4}{3} \right)$
Ex. MCQ | Q 30 | Page 118

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

•  4

•  2

• −2

• 0

Ex. MCQ | 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

Ex. MCQ | Q 32 | Page 119
$\int\limits_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$  equals to
Ex. MCQ | 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

Ex. MCQ | Q 34 | Page 120
$\int\limits_0^{\pi/2} x \sin x\ dx$  is equal to
•  π/4

•  π/2

• π

• 1

Ex. MCQ | Q 35 | Page 120
$\int\limits_0^{\pi/2} \sin\ 2x\ \log\ \tan x\ dx$  is equal to
• π

•  π/2

•  0

Ex. MCQ | Q 36 | Page 120

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

• π/4

• π/8

• π/2

• 0

Ex. MCQ | 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

• $- \frac{\pi}{2}\ln 2$

Ex. MCQ | 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

• $\int\limits_0^a f\left( x \right) dx + \int\limits_0^a f\left( 2a - x \right) dx$

• $\int\limits_0^a f\left( x \right) dx + \int\limits_0^{2a} f\left( 2a - x \right) dx$
Ex. MCQ | Q 39 | Page 120

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

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

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

• $\frac{b - a}{2} \int\limits_a^b f\left( x \right) dx$
• $\frac{b + a}{2} \int\limits_a^b f\left( x \right) dx$
Ex. MCQ | 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

Ex. MCQ | 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

Ex. MCQ | 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

#### Chapter 20: Definite Integrals Exercise Revision Exercise solutions [Pages 12 - 123]

Ex. Revision Exercise | Q 1 | Page 121

$\int\limits_0^4 x\sqrt{4 - x} dx$

Ex. Revision Exercise | Q 2 | Page 121

$\int\limits_1^2 x\sqrt{3x - 2} dx$

Ex. Revision Exercise | Q 3 | Page 121

$\int\limits_1^5 \frac{x}{\sqrt{2x - 1}} dx$

Ex. Revision Exercise | Q 4 | Page 121

$\int\limits_0^1 \cos^{- 1} x dx$

Ex. Revision Exercise | Q 5 | Page 121

$\int\limits_0^1 \tan^{- 1} x dx$

Ex. Revision Exercise | Q 6 | Page 121

$\int\limits_0^1 \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) dx$

Ex. Revision Exercise | Q 7 | Page 121

$\int\limits_0^1 \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) dx$

Ex. Revision Exercise | Q 8 | Page 121

$\int\limits_0^{1/\sqrt{3}} \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx$

Ex. Revision Exercise | Q 9 | Page 121

$\int\limits_0^1 \frac{1 - x}{1 + x} dx$

Ex. Revision Exercise | Q 10 | Page 121

$\int\limits_0^{\pi/3} \frac{\cos x}{3 + 4 \sin x} dx$

Ex. Revision Exercise | Q 11 | Page 121

$\int\limits_0^{\pi/2} \frac{\sin^2 x}{\left( 1 + \cos x \right)^2} dx$

Ex. Revision Exercise | Q 12 | Page 121

$\int\limits_0^{\pi/2} \frac{\sin x}{\sqrt{1 + \cos x}} dx$

Ex. Revision Exercise | Q 13 | Page 121

$\int\limits_0^{\pi/2} \frac{\cos x}{1 + \sin^2 x} dx$

Ex. Revision Exercise | Q 14 | Page 121

$\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx$

Ex. Revision Exercise | Q 15 | Page 121

$\int\limits_0^\infty \frac{x}{\left( 1 + x \right)\left( 1 + x^2 \right)} dx$

Ex. Revision Exercise | Q 16 | Page 121

$\int\limits_0^{\pi/4} \sin 2x \sin 3x dx$

Ex. Revision Exercise | Q 17 | Page 121

$\int\limits_0^1 \sqrt{\frac{1 - x}{1 + x}} dx$

Ex. Revision Exercise | Q 18 | Page 121

$\int\limits_1^2 \frac{1}{x^2} e^{- 1/x} dx$

Ex. Revision Exercise | Q 19 | Page 121

$\int\limits_0^{\pi/4} \cos^4 x \sin^3 x dx$

Ex. Revision Exercise | Q 20 | Page 121

$\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{5/2}} dx$

Ex. Revision Exercise | Q 21 | Page 121

$\int\limits_0^{\pi/2} x^2 \cos 2x dx$

Ex. Revision Exercise | Q 22 | Page 121

$\int\limits_0^1 \log\left( 1 + x \right) dx$

Ex. Revision Exercise | Q 23 | Page 121

Evaluate the following integrals :-

$\int_2^4 \frac{x^2 + x}{\sqrt{2x + 1}}dx$

Ex. Revision Exercise | Q 24 | Page 121

$\int\limits_0^1 x \left( \tan^{- 1} x \right)^2 dx$

Ex. Revision Exercise | Q 25 | Page 121

$\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx$

Ex. Revision Exercise | Q 26 | Page 121

$\int\limits_1^2 \frac{x + 3}{x\left( x + 2 \right)} dx$

Ex. Revision Exercise | Q 27 | Page 122

$\int\limits_0^{\pi/4} e^x \sin x dx$

Ex. Revision Exercise | Q 28 | Page 122

$\int\limits_0^{\pi/4} \tan^4 x dx$

Ex. Revision Exercise | Q 29 | Page 122

$\int\limits_0^1 \left| 2x - 1 \right| dx$

Ex. Revision Exercise | Q 31 | Page 122

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

Ex. Revision Exercise | Q 31 | Page 122

$\int\limits_0^{\pi/2} \left| \sin x - \cos x \right| dx$

Ex. Revision Exercise | Q 32 | Page 122

$\int\limits_0^1 \left| \sin 2\pi x \right| dx$

Ex. Revision Exercise | Q 33 | Page 122

$\int\limits_1^3 \left| x^2 - 4 \right| dx$

Ex. Revision Exercise | Q 34 | Page 122

$\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx$

Ex. Revision Exercise | Q 35 | Page 122

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

Ex. Revision Exercise | Q 36 | Page 122

$\int\limits_{- a}^a \frac{x e^{x^2}}{1 + x^2} dx$

Ex. Revision Exercise | Q 37 | Page 122

$\int\limits_0^{\pi/2} \frac{1}{1 + \cot^7 x} dx$

Ex. Revision Exercise | Q 38 | Page 122

$\int\limits_0^{2\pi} \cos^7 x dx$

Ex. Revision Exercise | Q 39 | Page 122

$\int\limits_0^a \frac{\sqrt{x}}{\sqrt{x} + \sqrt{a - x}} dx$

Ex. Revision Exercise | Q 40 | Page 122

$\int\limits_0^{\pi/2} \frac{1}{1 + \tan^3 x} dx$

Ex. Revision Exercise | Q 41 | Page 122

$\int\limits_0^\pi \frac{x \sin x}{1 + \cos^2 x} dx$

Ex. Revision Exercise | Q 42 | Page 122

$\int\limits_0^\pi x \sin x \cos^4 x dx$

Ex. Revision Exercise | Q 43 | Page 122

$\int\limits_0^\pi \frac{x}{a^2 \cos^2 x + b^2 \sin^2 x} dx$

Ex. Revision Exercise | Q 44 | Page 122

$\int\limits_{- \pi/4}^{\pi/4} \left| \tan x \right| dx$

Ex. Revision Exercise | Q 45 | Page 122

$\int\limits_0^{15} \left[ x^2 \right] dx$

Ex. Revision Exercise | Q 46 | Page 122

$\int\limits_0^\pi \frac{x}{1 + \cos \alpha \sin x} dx$

Ex. Revision Exercise | Q 47 | Page 12

$\int\limits_0^{\pi/2} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} dx$

Ex. Revision Exercise | Q 48 | Page 122

$\int\limits_0^{\pi/2} \frac{\cos^2 x}{\sin x + \cos x} dx$

Ex. Revision Exercise | Q 49 | Page 122

$\int\limits_0^\pi \cos 2x \log \sin x dx$

Ex. Revision Exercise | Q 50 | Page 122

$\int\limits_0^\pi \frac{x}{a^2 - \cos^2 x} dx, a > 1$

Ex. Revision Exercise | Q 51 | Page 122

$\int\limits_0^\pi \frac{x \tan x}{\sec x + \tan x} dx$

Ex. Revision Exercise | Q 52 | Page 122

$\int\limits_2^3 \frac{\sqrt{x}}{\sqrt{5 - x} + \sqrt{x}} dx$

Ex. Revision Exercise | Q 53 | Page 122

$\int\limits_0^{\pi/2} \frac{\sin^2 x}{\sin x + \cos x} dx$

Ex. Revision Exercise | Q 54 | Page 122

$\int\limits_0^{\pi/2} \frac{x}{\sin^2 x + \cos^2 x} dx$

Ex. Revision Exercise | Q 55 | Page 122

$\int\limits_{- \pi}^\pi x^{10} \sin^7 x dx$

Ex. Revision Exercise | Q 56 | Page 122

$\int\limits_0^1 \cot^{- 1} \left( 1 - x + x^2 \right) dx$

Ex. Revision Exercise | Q 57 | Page 122

$\int\limits_0^\pi \frac{dx}{6 - \cos x}dx$

Ex. Revision Exercise | Q 58 | Page 122

$\int\limits_0^{\pi/2} \frac{1}{2 \cos x + 4 \sin x} dx$

Ex. Revision Exercise | Q 59 | Page 123

$\int\limits_{\pi/6}^{\pi/2} \frac{\ cosec x \cot x}{1 + {cosec}^2 x} dx$

Ex. Revision Exercise | Q 60 | Page 123

$\int\limits_0^{\pi/2} \frac{dx}{4 \cos x + 2 \sin x}dx$

Ex. Revision Exercise | Q 61 | Page 123

$\int\limits_0^4 x dx$

Ex. Revision Exercise | Q 62 | Page 123

$\int\limits_0^2 \left( 2 x^2 + 3 \right) dx$

Ex. Revision Exercise | Q 63 | Page 123

$\int\limits_1^4 \left( x^2 + x \right) dx$

Ex. Revision Exercise | Q 64 | Page 123

$\int\limits_{- 1}^1 e^{2x} dx$

Ex. Revision Exercise | Q 65 | Page 123

$\int\limits_2^3 e^{- x} dx$

Ex. Revision Exercise | Q 66 | Page 123

$\int\limits_1^3 \left( 2 x^2 + 5x \right) dx$

Ex. Revision Exercise | Q 67 | Page 123

$\int\limits_1^3 \left( x^2 + 3x \right) dx$

Ex. Revision Exercise | Q 68 | Page 123

$\int\limits_0^2 \left( x^2 + 2 \right) dx$

Ex. Revision Exercise | Q 69 | Page 123

$\int\limits_0^3 \left( x^2 + 1 \right) dx$

## Chapter 20: Definite Integrals

Ex. 20.1Ex. 20.2Ex. 20.3Ex. 20.4Ex. 20.5Ex. 20.6Ex. Very Short AnswersEx. MCQEx. Revision Exercise

## RD Sharma solutions for Class 12 Mathematics chapter 20 - Definite Integrals

RD Sharma solutions for Class 12 Maths chapter 20 (Definite Integrals) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 20 Definite Integrals are Integration as an Inverse Process of Differentiation, Methods of Integration - Integration by Substitution, Methods of Integration - Integration Using Partial Fractions, Methods of Integration - Integration by Parts, Evaluation of Simple Integrals of the Following Types and Problems, Definite Integral as the Limit of a Sum, Fundamental Theorem of Calculus, Properties of Definite Integrals, Evaluation of Definite Integrals by Substitution, Introduction of Integrals, Integration Using Trigonometric Identities, Properties of Indefinite Integral, Indefinite Integral by Inspection, Integrals of Some Particular Functions, Geometrical Interpretation of Indefinite Integral, Comparison Between Differentiation and Integration, Indefinite Integral Problems, Definite Integrals Problems.

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