# RD Sharma solutions for Class 12 Maths chapter 20 - Definite Integrals [Latest edition]

## Solutions for Chapter 20: Definite Integrals

Below listed, you can find solutions for Chapter 20 of CBSE, Karnataka Board PUC RD Sharma for Class 12 Maths.

Exercise 20.1Exercise 20.2Exercise 20.3Exercise 20.4Exercise 20.5Exercise 20.6Very Short AnswersMCQRevision Exercise
Exercise 20.1 [Pages 16 - 18]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.1 [Pages 16 - 18]

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

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

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

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

Evaluate the following definite integrals:

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

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

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

Evaluate the following definite integral:

$\int_0^1 \frac{1}{\sqrt{\left( x - 1 \right)\left( 2 - x \right)}}dx$
Exercise 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.

Exercise 20.1 | Q 61 | Page 18

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

Exercise 20.1 | Q 62 | Page 18
$\int_\pi^\frac{3\pi}{2} \sqrt{1 - \cos2x}dx$
Exercise 20.1 | Q 63 | Page 18
$\int_0^{2\pi} \sqrt{1 + \sin\frac{x}{2}}dx$
Exercise 20.1 | Q 64 | Page 18
$\int_0^\frac{\pi}{4} \left( \tan x + \cot x \right)^{- 2} dx$
Exercise 20.1 | Q 65 | Page 18
$\int_0^1 x\log\left( 1 + 2x \right)dx$
Exercise 20.1 | Q 66 | Page 18
$\int_\frac{\pi}{6}^\frac{\pi}{3} \left( \tan x + \cot x \right)^2 dx$
Exercise 20.1 | Q 67 | Page 18
$\int_0^\frac{\pi}{4} \left( a^2 \cos^2 x + b^2 \sin^2 x \right)dx$
Exercise 20.1 | Q 68 | Page 18
$\int_0^1 \frac{1}{1 + 2x + 2 x^2 + 2 x^3 + x^4}dx$
Exercise 20.2 [Pages 38 - 40]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.2 [Pages 38 - 40]

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

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

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

Exercise 20.2 | Q 45 | Page 40
$\int\limits_1^2 \frac{1}{x \left( 1 + \log x \right)^2} dx$
Exercise 20.2 | Q 46 | Page 40
$\int\limits_0^{\pi/2} \cos^5 x\ dx$
Exercise 20.2 | Q 47 | Page 40
$\int\limits_4^9 \frac{\sqrt{x}}{\left( 30 - x^{3/2} \right)^2} dx$
Exercise 20.2 | Q 48 | Page 40
$\int\limits_0^\pi \sin^3 x\left( 1 + 2 \cos x \right) \left( 1 + \cos x \right)^2 dx$
Exercise 20.2 | Q 49 | Page 40
$\int\limits_0^{\pi/2} 2 \sin x \cos x \tan^{- 1} \left( \sin x \right) dx$
Exercise 20.2 | Q 50 | Page 39
$\int\limits_0^{\pi/2} \sin 2x \tan^{- 1} \left( \sin x \right) dx$
Exercise 20.2 | Q 51 | Page 40
$\int\limits_0^1 \left( \cos^{- 1} x \right)^2 dx$
Exercise 20.2 | Q 52 | Page 40
$\int\limits_0^a \sin^{- 1} \sqrt{\frac{x}{a + x}} dx$
Exercise 20.2 | Q 53 | Page 40
$\int\limits_{\pi/3}^{\pi/2} \frac{\sqrt{1 + \cos x}}{\left( 1 - \cos x \right)^{3/2}} dx$
Exercise 20.2 | Q 54 | Page 40
$\int\limits_0^a x \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} dx$
Exercise 20.2 | Q 55 | Page 40
$\int\limits_{- a}^a \sqrt{\frac{a - x}{a + x}} dx$
Exercise 20.2 | Q 56 | Page 40
$\int\limits_0^{\pi/2} \frac{\sin x \cos x}{\cos^2 x + 3 \cos x + 2} dx$
Exercise 20.2 | Q 57 | Page 40
$\int_0^\frac{\pi}{2} \frac{\tan x}{1 + m^2 \tan^2 x}dx$
Exercise 20.2 | Q 58 | Page 40
$\int_0^\frac{1}{2} \frac{1}{\left( 1 + x^2 \right)\sqrt{1 - x^2}}dx$
Exercise 20.2 | Q 59 | Page 40
$\int_\frac{1}{3}^1 \frac{\left( x - x^3 \right)^\frac{1}{3}}{x^4}dx$
Exercise 20.2 | Q 60 | Page 40
$\int_0^\frac{\pi}{4} \frac{\sin^2 x \cos^2 x}{\left( \sin^3 x + \cos^3 x \right)^2}dx$
Exercise 20.2 | Q 61 | Page 40
$\int_0^\frac{\pi}{2} \sqrt{\cos x - \cos^3 x}\left( \sec^2 x - 1 \right) \cos^2 xdx$
Exercise 20.2 | Q 62 | Page 40
$\int_0^\frac{\pi}{2} \frac{\cos x}{\left( \cos\frac{x}{2} + \sin\frac{x}{2} \right)^n}dx$
Exercise 20.3 [Pages 55 - 56]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.3 [Pages 55 - 56]

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

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

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

Exercise 20.3 | Q 2 | Page 56

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Exercise 20.3 | Q 7 | Page 56

Evaluate the following integral:

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

Exercise 20.3 | Q 8 | Page 56

Evaluate the following integral:

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

Exercise 20.3 | Q 9 | Page 56

Evaluate the following integral:

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

Exercise 20.3 | Q 10 | Page 56

Evaluate the following integral:

$\int\limits_1^2 \left| x - 3 \right| dx$
Exercise 20.3 | Q 11 | Page 56

Evaluate the following integral:

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

Evaluate the following integral:

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

Exercise 20.3 | Q 13 | Page 56

Evaluate the following integral:

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

Evaluate the following integral:

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

Exercise 20.3 | Q 15 | Page 56

Evaluate the following integral:

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

Exercise 20.3 | Q 16 | Page 56

Evaluate the following integral:

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

Evaluate the following integral:

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

Exercise 20.3 | Q 18 | Page 56

Evaluate the following integral:

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

Exercise 20.3 | Q 19 | Page 56

Evaluate the following integral:

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

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

Exercise 20.3 | Q 23 | Page 56
$\int_0^\pi \cos x\left| \cos x \right|dx$
Exercise 20.3 | Q 24 | Page 56
$\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \left( 2\sin\left| x \right| + \cos\left| x \right| \right)dx$
Exercise 20.3 | Q 25 | Page 56
$\int_{- \frac{\pi}{2}}^\pi \sin^{- 1} \left( \sin x \right)dx$
Exercise 20.3 | Q 26 | Page 56
$\int_{- \frac{\pi}{2}}^\frac{\pi}{2} \frac{- \frac{\pi}{2}}{\sqrt{\cos x \sin^2 x}}dx$
Exercise 20.3 | Q 27 | Page 56
$\int_0^2 2x\left[ x \right]dx$
Exercise 20.3 | Q 28 | Page 56
$\int_0^{2\pi} \cos^{- 1} \left( \cos x \right)dx$
Exercise 20.4 [Page 61]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.4 [Page 61]

Exercise 20.4 | Q 1 | Page 61

Evaluate each of the following integral:

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

Exercise 20.4 | Q 2 | Page 61

Evaluate each of the following integral:

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

Exercise 20.4 | Q 3 | Page 61

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

Exercise 20.4 | Q 5 | Page 61

Evaluate each of the following integral:

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

Exercise 20.4 | Q 6 | Page 61

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

Exercise 20.4 | Q 8 | Page 61

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

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

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

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

Exercise 20.5 [Pages 94 - 96]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.5 [Pages 94 - 96]

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

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

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

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

Evaluate the following integral:

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

Exercise 20.5 | Q 19 | Page 95

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Evaluate the following integral:

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

Exercise 20.5 | Q 35 | Page 95

Evaluate the following integral:

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

Exercise 20.5 | Q 36 | Page 95

Evaluate the following integral:

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

Evaluate

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

Exercise 20.5 | Q 38 | Page 95

Evaluate the following integral:

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

Exercise 20.5 | Q 39 | Page 95

Evaluate the following integral:

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

Exercise 20.5 | Q 40 | Page 95

int_0^(2a)f(x)dx

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

Evaluate :

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

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

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

Exercise 20.5 | Q 44 | Page 96

If f(2a − x) = −f(x), prove that

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

If f is an integrable function, show that

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

Exercise 20.5 | Q 45.2 | Page 96

If f is an integrable function, show that

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

Exercise 20.5 | Q 46 | Page 96

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

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

Exercise 20.5 | Q 47 | Page 96

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

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

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

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

Prove that:

$\int_0^\pi xf\left( \sin x \right)dx = \frac{\pi}{2} \int_0^\pi f\left( \sin x \right)dx$
Exercise 20.6 [Pages 110 - 111]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Exercise 20.6 [Pages 110 - 111]

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

Evaluate the following integrals as limit of sums:

$\int_1^3 \left( 3 x^2 + 1 \right)dx$
Very Short Answers [Pages 115 - 116]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Very Short Answers [Pages 115 - 116]

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

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

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

Evaluate each of the following integral:

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

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

Evaluate each of the following  integral:

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

Very Short Answers | Q 27 | Page 115

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

Evaluate each of the following integral:

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

Very Short Answers | Q 30 | Page 115

Solve each of the following integral:

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

Very Short Answers | Q 32 | Page 116

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

Very Short Answers | Q 33 | Page 116

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

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.

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.
Very Short Answers | Q 36 | Page 116

Evaluate :

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

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

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

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

MCQ [Pages 117 - 120]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals MCQ [Pages 117 - 120]

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

MCQ | Q 2 | Page 117

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

• 0

• 1/2

• 2

• 3/2

MCQ | Q 3 | Page 117

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

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

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

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

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

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

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
• $\log\left( \frac{2}{3} \right)$
• $\log\left( \frac{3}{2} \right)$
• $\log\left( \frac{3}{4} \right)$
• $\log\left( \frac{4}{3} \right)$
MCQ | Q 9 | Page 117

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

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

• $2\sqrt{3} \tan^{- 1} \sqrt{3}$
MCQ | Q 10 | Page 117

int_0^1 sqrt((1 - "x")/(1 + "x")) "dx"

• $\frac{\pi}{2}$
• $\frac{\pi}{2} - 1$

• $\frac{\pi}{2} + 1$
•  π + 1

• None of these

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

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

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

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

•  e − 1

• e + 1

•  0

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

MCQ | Q 17 | Page 118

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

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

• 2

• − 1

• − 2

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

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

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

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

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

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)

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

• 2

• 0

• 4

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)

• $\frac{3 x^2}{\ln x}$
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

• $9 \left( \frac{\pi}{2} \right)^9$
• $10 \left( \frac{\pi}{2} \right)^9$
• $\left( \frac{\pi}{2} \right)^9$
• $9 \left( \frac{\pi}{2} \right)^8$
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

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
• $\ln\left( \frac{1}{3} \right)$
• $\ln\left( \frac{2}{3} \right)$
• $\ln\left( \frac{3}{2} \right)$
• $\ln\left( \frac{4}{3} \right)$
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

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

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

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

•  π/2

• π

• 1

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

•  π/2

•  0

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

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

• $- \frac{\pi}{2}\ln 2$

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

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

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

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

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

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

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

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

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

Revision Exercise [Pages 121 - 123]

### RD Sharma solutions for Class 12 Maths Chapter 20 Definite Integrals Revision Exercise [Pages 121 - 123]

Revision Exercise | Q 1 | Page 121

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

Revision Exercise | Q 2 | Page 121

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

Revision Exercise | Q 3 | Page 121

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

Revision Exercise | Q 4 | Page 121

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

Revision Exercise | Q 5 | Page 121

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

Revision Exercise | Q 6 | Page 121

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

Revision Exercise | Q 7 | Page 121

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

Revision Exercise | Q 8 | Page 121

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

Revision Exercise | Q 9 | Page 121

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

Revision Exercise | Q 10 | Page 121

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

Revision Exercise | Q 11 | Page 121

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

Revision Exercise | Q 12 | Page 121

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

Revision Exercise | Q 13 | Page 121

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

Revision Exercise | Q 14 | Page 121

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

Revision Exercise | Q 15 | Page 121

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

Revision Exercise | Q 16 | Page 121

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

Revision Exercise | Q 17 | Page 121

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

Revision Exercise | Q 18 | Page 121

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

Revision Exercise | Q 19 | Page 121

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

Revision Exercise | Q 20 | Page 121

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

Revision Exercise | Q 21 | Page 121

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

Revision Exercise | Q 22 | Page 121

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

Revision Exercise | Q 23 | Page 121

Evaluate the following integrals :-

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

Revision Exercise | Q 24 | Page 121

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

Revision Exercise | Q 25 | Page 121

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

Revision Exercise | Q 26 | Page 121

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

Revision Exercise | Q 27 | Page 122

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

Revision Exercise | Q 28 | Page 122

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

Revision Exercise | Q 29 | Page 122

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

Revision Exercise | Q 30 | Page 122

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

Revision Exercise | Q 31 | Page 122

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

Revision Exercise | Q 32 | Page 122

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

Revision Exercise | Q 33 | Page 122

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

Revision Exercise | Q 34 | Page 122

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

Revision Exercise | Q 35 | Page 122

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

Revision Exercise | Q 36 | Page 122

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

Revision Exercise | Q 37 | Page 122

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

Revision Exercise | Q 38 | Page 122

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

Revision Exercise | Q 39 | Page 122

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

Revision Exercise | Q 40 | Page 122

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

Revision Exercise | Q 41 | Page 122

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

Revision Exercise | Q 42 | Page 122

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

Revision Exercise | Q 43 | Page 122

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

Revision Exercise | Q 44 | Page 122

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

Revision Exercise | Q 45 | Page 122

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

Revision Exercise | Q 46 | Page 122

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

Revision Exercise | Q 47 | Page 12

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

Revision Exercise | Q 48 | Page 122

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

Revision Exercise | Q 49 | Page 122

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

Revision Exercise | Q 50 | Page 122

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

Revision Exercise | Q 51 | Page 122

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

Revision Exercise | Q 52 | Page 122

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

Revision Exercise | Q 53 | Page 122

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

Revision Exercise | Q 54 | Page 122

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

Revision Exercise | Q 55 | Page 122

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

Revision Exercise | Q 56 | Page 122

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

Revision Exercise | Q 57 | Page 122

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

Revision Exercise | Q 58 | Page 122

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

Revision Exercise | Q 59 | Page 123

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

Revision Exercise | Q 60 | Page 123

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

Revision Exercise | Q 61 | Page 123

$\int\limits_0^4 x dx$

Revision Exercise | Q 62 | Page 123

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

Revision Exercise | Q 63 | Page 123

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

Revision Exercise | Q 64 | Page 123

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

Revision Exercise | Q 65 | Page 123

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

Revision Exercise | Q 66 | Page 123

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

Revision Exercise | Q 67 | Page 123

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

Revision Exercise | Q 68 | Page 123

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

Revision Exercise | Q 69 | Page 123

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

## Solutions for Chapter 20: Definite Integrals

Exercise 20.1Exercise 20.2Exercise 20.3Exercise 20.4Exercise 20.5Exercise 20.6Very Short AnswersMCQRevision Exercise

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

Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Class 12 Maths CBSE, Karnataka Board PUC solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. RD Sharma solutions for Mathematics Class 12 Maths CBSE, Karnataka Board PUC 20 (Definite Integrals) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.

Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.

Concepts covered in Class 12 Maths chapter 20 Definite Integrals are Definite Integrals Problems, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Integrals of Some Particular Functions, Indefinite Integral by Inspection, Some 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, Geometrical Interpretation of Indefinite Integrals.

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

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