Solutions for Chapter 19: Indefinite Integrals
Below listed, you can find solutions for Chapter 19 of CBSE, Karnataka Board PUC RD Sharma for Mathematics [English] Class 12.
Exercise 19.01Exercise 19.02Exercise 19.03Exercise 19.04Exercise 19.05Exercise 19.06Exercise 19.07Exercise 19.08Exercise 19.09Exercise 19.10Exercise 19.11Exercise 19.12Exercise 19.13Exercise 19.14Exercise 19.15Exercise 19.16Exercise 19.17Exercise 19.18Exercise 19.19Exercise 19.2Exercise 19.21Exercise 19.22Exercise 19.23Exercise 19.24Exercise 19.25Exercise 19.26Exercise 19.27Exercise 19.28Exercise 19.29Exercise 19.30Exercise 19.31Exercise 19.32Very Short AnswersMCQRevision Excercise
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.01 [Page 4]
1.1Page 4
Evaluate of the following integral:
(i) \[\int x^4 dx\]
1.2Page 4
Evaluate of the following integral:
\[\int x^\frac{5}{4} dx\]
1.3Page 4
Evaluate of the following integral:
1.4Page 4
Evaluate of the following integral:
\[\int\frac{1}{x^{3/2}}dx\]
1.5Page 4
Evaluate of the following integral:
1.6Page 4
Evaluate of the following integral:
\[\int\frac{1}{\sqrt[3]{x^2}}dx\]
1.7Page 4
Evaluate of the following integral:
\[\int 3^{2 \log_3} {}^x dx\]
1.8Page 4
Evaluate of the following integral:
\[\int \log_x \text{x dx}\]
2.1Page 4
Evaluate:
\[\int\sqrt{\frac{1 + \cos 2x}{2}}dx\]
2.2Page 4
Evaluate:
\[\int\sqrt{\frac{1 - \cos 2x}{2}}dx\]
3Page 4
Evaluate :
\[\int\frac{e^{6 \log_e x} - e^{5 \log_e x}}{e^{4 \log_e x} - e^{3 \log_e x}}dx\]
4Page 4
Evaluate:
\[\int\frac{1}{a^x b^x}dx\]
5.1Page 4
Evaluate:
\[\int\frac{\cos 2x + 2 \sin^2 x}{\sin^2 x}dx\]
5.2Page 4
Evaluate:
\[\int\frac{2 \cos^2 x - \cos 2x}{\cos^2 x}dx\]
6Page 4
Evaluate:
\[\int\frac{e\log \sqrt{x}}{x}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.02 [Pages 14 - 15]
1Page 14
\[\int\left( 3x\sqrt{x} + 4\sqrt{x} + 5 \right)dx\]
2Page 14
\[\int\left( 2^x + \frac{5}{x} - \frac{1}{x^{1/3}} \right)dx\]
3Page 14
`int{sqrtx(ax^2+bx+c)}dx`
4Page 14
\[\int\left( 2 - 3x \right) \left( 3 + 2x \right) \left( 1 - 2x \right) dx\]
5Page 14
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
6Page 14
\[\int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 dx\]
7Page 14
\[\int\frac{\left( 1 + x \right)^3}{\sqrt{x}} dx\]
8Page 14
\[\int\left\{ x^2 + e^{\log x}+ \left( \frac{e}{2} \right)^x \right\} dx\]
9Page 14
\[\int\left( x^e + e^x + e^e \right) dx\]
10Page 14
\[\int\sqrt{x}\left( x^3 - \frac{2}{x} \right) dx\]
11Page 14
\[\int\frac{1}{\sqrt{x}}\left( 1 + \frac{1}{x} \right) dx\]
12Page 14
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
13Page 14
\[\int\frac{x^{- 1/3} + \sqrt{x} + 2}{\sqrt[3]{x}} dx\]
14Page 14
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
15Page 15
\[\int\sqrt{x}\left( 3 - 5x \right) dx\]
16Page 15
\[\int\frac{\left( x + 1 \right)\left( x - 2 \right)}{\sqrt{x}} dx\]
17Page 15
\[\int\frac{x^5 + x^{- 2} + 2}{x^2} dx\]
18Page 15
\[\int \left( 3x + 4 \right)^2 dx\]
19Page 15
\[\int\frac{2 x^4 + 7 x^3 + 6 x^2}{x^2 + 2x} dx\]
20Page 15
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
21Page 15
\[\int\frac{\sin^2 x}{1 + \cos x} \text{dx} \]
22Page 15
\[\int\left( \sec^2 x + {cosec}^2 x \right) dx\]
23Page 15
\[\int\frac{\sin^3 x - \cos^3 x}{\sin^2 x \cos^2 x} dx\]
24Page 15
\[\int\frac{5 \cos^3 x + 6 \sin^3 x}{2 \sin^2 x \cos^2 x} dx\]
25Page 15
\[\int \left( \tan x + \cot x \right)^2 dx\]
26Page 15
\[\int\frac{1 - \cos 2x}{1 + \cos 2x} dx\]
27Page 15
\[\int\frac{\cos x}{1 - \cos x} \text{dx }or \int\frac{\cot x}{\text{cosec } {x }- \cot x} dx\]
28Page 15
\[\int\frac{\cos^2 x - \sin^2 x}{\sqrt{1} + \cos 4x} dx\]
29Page 15
\[\int\frac{1}{1 - \cos x} dx\]
30Page 15
\[\int\frac{1}{1 - \sin x} dx\]
31Page 15
\[\int\frac{\tan x}{\sec x + \tan x} dx\]
32Page 15
` ∫ {cosec x} / {"cosec x "- cot x} ` dx
33Page 15
\[\int\frac{1}{1 + \cos 2x} dx\]
34Page 15
\[\int\frac{1}{1 - \cos 2x} dx\]
35Page 15
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
36Page 15
\[\int \cos^{- 1} \left( \sin x \right) dx\]
37Page 15
\[\int \cot^{- 1} \left( \frac{\sin 2x}{1 - \cos 2x} \right) dx\]
38Page 15
\[\int \sin^{- 1} \left( \frac{2 \tan x}{1 + \tan^2 x} \right) dx\]
39Page 15
\[\int\frac{\left( x^3 + 8 \right)\left( x - 1 \right)}{x^2 - 2x + 4} dx\]
40Page 15
\[\int \left( a \tan x + b \cot x \right)^2 dx\]
41Page 15
\[\int\frac{x^3 - 3 x^2 + 5x - 7 + x^2 a^x}{2 x^2} dx\]
42Page 15
\[\int\frac{\cos x}{1 + \cos x} dx\]
43Page 15
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
45Page 15
If f' (x) = x − \[\frac{1}{x^2}\] and f (1) \[\frac{1}{2}, find f(x)\]
46Page 15
If f' (x) = x + b, f(1) = 5, f(2) = 13, find f(x)
47Page 15
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
48Page 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)
49Page 15
Write the primitive or anti-derivative of
\[f\left( x \right) = \sqrt{x} + \frac{1}{\sqrt{x}} .\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.03 [Pages 23 - 24]
1Page 23
\[\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} \text{dx} \]
2Page 23
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
3Page 23
\[\int\frac{1}{2 - 3x} + \frac{1}{\sqrt{3x - 2}} dx\]
4Page 23
\[\int\frac{x + 3}{\left( x + 1 \right)^4} dx\]
5Page 23
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
6Page 23
\[\int\frac{1}{\sqrt{2x + 3} + \sqrt{2x - 3}} dx\]
7Page 23
\[\int\frac{2x}{\left( 2x + 1 \right)^2} dx\]
8Page 23
\[\int\frac{1}{\sqrt{x + a} + \sqrt{x + b}} dx\]
9Page 23
\[\int\sin x\sqrt{1 + \cos 2x} dx\]
10Page 23
\[\int\frac{1 + \cos x}{1 - \cos x} dx\]
11Page 23
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
12Page 23
\[\int\frac{1}{1 - \sin\frac{x}{2}} dx\]
14Page 23
\[\int \left( e^x + 1 \right)^2 e^x dx\]
15Page 23
\[\int \left( e^x + 1 \right)^2 e^x dx\]
16Page 23
\[\int\frac{1 + \cos 4x}{\cot x - \tan x} dx\]
17Page 23
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
18Page 23
\[\int \tan^2 \left( 2x - 3 \right) dx\]
19Page 24
\[\int\frac{1}{\text{cos}^2\text{ x }\left( 1 - \text{tan x} \right)^2} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.04 [Page 30]
1Page 30
\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]
2Page 30
\[\int\frac{x^3}{x - 2} dx\]
3Page 30
\[\int\frac{x^2 + x + 5}{3x + 2} dx\]
4Page 30
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
5Page 30
\[\int\frac{x^2 + 3x - 1}{\left( x + 1 \right)^2} dx\]
6Page 30
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.05 [Page 33]
1Page 33
\[\int\frac{x + 1}{\sqrt{2x + 3}} dx\]
3Page 33
\[\int\frac{x - 1}{\sqrt{x + 4}} dx\]
4Page 33
\[\int\left( x + 2 \right) \sqrt{3x + 5} \text{dx} \]
5Page 33
\[\int\frac{2x + 1}{\sqrt{3x + 2}} dx\]
6Page 33
\[\int\frac{3x + 5}{\sqrt{7x + 9}} dx\]
7Page 33
\[\int\frac{x}{\sqrt{x + 4}} dx\]
8Page 33
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
9Page 33
\[\int\left( 5x + 3 \right) \sqrt{2x - 1} dx\]
10Page 33
\[\int\frac{x}{\sqrt{x + a} - \sqrt{x + b}}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.06 [Page 36]
1Page 36
\[\int \text{sin}^2 \left( 2x + 5 \right) \text{dx}\]
2Page 36
\[\int \sin^3 \left( 2x + 1 \right) \text{dx}\]
4Page 36
\[\int \sin^2\text{ b x dx}\]
5Page 36
\[\int \sin^2 \frac{x}{2} dx\]
6Page 3
\[\int \cos^2 \frac{x}{2} dx\]
7Page 36
\[\int \cos^2 \text{nx dx}\]
8Page 36
` ∫ sin x \sqrt (1-cos 2x) dx `
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.07 [Page 38]
4Page 38
\[\int\text{sin mx }\text{cos nx dx m }\neq n\]
5Page 38
Integrate the following integrals:
\[\int\text{sin 2x sin 4x sin 6x dx} \]
6Page 38
Integrate the following integrals:
\[\int\text { sin x cos 2x sin 3x dx}\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.08 [Pages 47 - 48]
1Page 47
\[\int\frac{1}{\sqrt{1 - \cos 2x}} dx\]
2Page 47
\[\int\frac{1}{\sqrt{1 + \cos x}} dx\]
3Page 47
\[\int\sqrt{\frac{1 + \cos 2x}{1 - \cos 2x}} dx\]
4Page 47
\[\int\sqrt{\frac{1 - \cos x}{1 + \cos x}} dx\]
5Page 47
Evaluate the following integrals:
`int "sec x"/"sec 2x" "dx"`
6Page 47
\[\int\frac{\cos 2x}{\left( \cos x + \sin x \right)^2} dx\]
7Page 47
\[\int\frac{\text{sin} \left( x - a \right)}{\text{sin}\left( x - b \right)} dx\]
8Page 47
\[\int\frac{\text{sin} \left( x - \alpha \right)}{\text{sin }\left( x + \alpha \right)} dx\]
9Page 47
\[\int\frac{1 + \tan x}{1 - \tan x} dx\]
10Page 47
\[\int\frac{\cos x}{\cos \left( x - a \right)} dx\]
11Page 47
\[\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx\]
12Page 47
\[\int\frac{e^{3x}}{e^{3x} + 1} dx\]
13Page 47
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
14Page 47
\[\int\frac{1 - \cot x}{1 + \cot x} dx\]
15Page 47
` ∫ {sec x "cosec " x}/{log ( tan x) }` dx
16Page 47
\[\int\frac{1}{x (3 + \log x)} dx\]
17Page 47
\[\int\frac{e^x + 1}{e^x + x} dx\]
18Page 47
\[\int\frac{1}{x \log x} dx\]
19Page 47
` ∫ {sin 2x} /{a cos^2 x + b sin^2 x } ` dx
20Page 47
\[\int\frac{\cos x}{2 + 3 \sin x} dx\]
21Page 47
\[\int\frac{1 - \sin x}{x + \cos x} dx\]
22Page 47
\[\int\frac{a}{b + c e^x} dx\]
23Page 47
\[\int\frac{1}{e^x + 1} dx\]
24Page 47
` ∫ {cot x}/ { log sin x} dx `
25Page 47
\[\int\frac{e^{2x}}{e^{2x} - 2} dx\]
26Page 47
\[\int\frac{2 \cos x - 3 \sin x}{6 \cos x + 4 \sin x} dx\]
27Page 48
\[\int\frac{\cos 2x + x + 1}{x^2 + \sin 2x + 2x} dx\]
28Page 48
\[\int\frac{1}{\cos\left( x + a \right) \cos\left( x + b \right)}dx\]
29Page 48
\[\int\frac{- \sin x + 2 \cos x}{2 \sin x + \cos x} dx\]
30Page 48
\[\int\frac{\cos 4x - \cos 2x}{\sin 4x - \sin 2x} dx\]
31Page 48
\[\int\frac{sec x}{\log \left( \text{sec x }+ \text{tan x} \right)} dx\]
32Page 48
` ∫ {"cosec" x }/ { log tan x/2 ` dx
33Page 48
\[\int\frac{1}{ x \text{log x } \text{log }\left( \text{log x }\right)} dx\]
34Page 48
\[\int\frac{{cosec}^2 x}{1 + \cot x} dx\]
35Page 48
\[\int\frac{10 x^9 + {10}^x \log_e 10}{{10}^x + x^{10}} dx\]
36Page 48
\[\int\frac{1 - \sin 2x}{x + \cos^2 x} dx\]
37Page 48
` ∫ {1+tan}/{ x + log sec x dx} `
38Page 48
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x} dx\]
39Page 48
\[\int\frac{x + 1}{x \left( x + \log x \right)} dx\]
40Page 48
\[\int\frac{1}{\sqrt{1 - x^2}\left( 2 + 3 \sin^{- 1} x \right)} dx\]
41Page 48
\[\int\frac{\sec^2 x}{\tan x + 2} dx\]
42Page 48
\[\int\frac{2 \cos 2x + \sec^2 x}{\sin 2x + \tan x - 5} dx\]
43Page 48
\[\int\frac{\sin 2x}{\sin 5x \sin 3x} dx\]
44Page 48
\[\int\frac{1 + \cot x}{x + \log \sin x} dx\]
45Page 48
\[\int\frac{1}{\sqrt{x}\left( \sqrt{x} + 1 \right)} dx\]
46Page 48
` ∫ tan 2x tan 3x tan 5x dx `
47Page 48
\[\int\left\{ 1 + \tan x \tan \left( x + \theta \right) \right\} dx\]
48Page 48
\[\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx\]
49Page 48
\[\int\frac{e^{x - 1} + x^{e - 1}}{e^x + x^e} dx\]
50Page 48
\[\int\frac{1}{\sin x \cos^2 x} dx\]
51Page 48
\[\int\frac{1}{\cos 3x - \cos x} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.09 [Pages 57 - 59]
2Page 57
\[\int\frac{\log\left( 1 + \frac{1}{x} \right)}{x \left( 1 + x \right)} dx\]
3Page 57
\[\int\frac{\left( 1 + \sqrt{x} \right)^2}{\sqrt{x}} dx\]
4Page 57
\[\int\sqrt{1 + e^x} . e^x dx\]
5Page 57
` = ∫ root (3){ cos^2 x} sin x dx `
6Page 57
\[\int\frac{e^x}{\left( 1 + e^x \right)^2} dx\]
7Page 57
` ∫ cot^3 x "cosec"^2 x dx `
8Page 57
\[\int\frac{\left\{ e^{\sin^{- 1} }x \right\}^2}{\sqrt{1 - x^2}} dx\]
9Page 57
\[\int\frac{1 + \sin x}{\sqrt{x - \cos x}} dx\]
10Page 57
\[\int\frac{1}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2} dx\]
11Page 58
\[\int\frac{\cot x}{\sqrt{\sin x}} dx\]
12Page 58
\[\int\frac{\tan x}{\sqrt{\cos x}} dx\]
13Page 58
\[\int\frac{\cos^3 x}{\sqrt{\sin x}} dx\]
14Page 58
\[\int\frac{\sin^3 x}{\sqrt{\cos x}} dx\]
15Page 58
\[\int\frac{1}{\sqrt{\tan^{- 1} x} . \left( 1 + x^2 \right)} dx\]
16Page 58
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
17Page 58
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
18Page 58
\[\int \sin^5\text{ x }\text{cos x dx}\]
19Page 58
\[\int \tan^{3/2} x \sec^2 \text{x dx}\]
20Page 58
\[\int\frac{x^3}{\left( x^2 + 1 \right)^3} dx\]
21Page 58
\[\int\left( 4x + 2 \right)\sqrt{x^2 + x + 1} \text{dx}\]
22Page 58
\[\int\frac{4x + 3}{\sqrt{2 x^2 + 3x + 1}} dx\]
23Page 58
\[\int\frac{1}{1 + \sqrt{x}} dx\]
24Page 58
\[\int e^{cos^2 x} \text{sin 2x dx}\]
25Page 58
\[\int\frac{1 + \cos x}{\left( x + \sin x \right)^3} dx\]
26Page 58
\[\int\frac{\cos x - \sin x}{1 + \sin 2x} dx\]
27Page 58
\[\int\frac{\sin 2x}{\left( a + b \cos 2x \right)^2} dx\]
28Page 58
\[\int\frac{\log x^2}{x} dx\]
29Page 58
\[\int\frac{\sin x}{\left( 1 + \cos x \right)^2} dx\]
30Page 58
\[\int\cot x \cdot \log \text{sin x dx}\]
31Page 58
\[\int\sec x \cdot \text{log} \left( \sec x + \tan x \right) dx\]
32Page 58
\[\text{ ∫ cosec x log} \left( \text{cosec x} - \cot x \right) dx\]
35Page 58
\[\int\frac{x \sin^{- 1} x^2}{\sqrt{1 - x^4}} dx\]
36Page 58
\[\int x^3 \sin \left( x^4 + 1 \right) dx\]
37Page 58
\[\int\frac{\left( x + 1 \right) e^x}{\cos^2 \left( x e^x \right)} dx\]
38Page 58
\[\int x^2 e^{x^3} \cos \left( e^{x^3} \right) dx\]
39Page 58
\[\int2x \sec^3 \left( x^2 + 3 \right) \tan \left( x^2 + 3 \right) dx\]
40Page 58
\[\int\left( \frac{x + 1}{x} \right) \left( x + \log x \right)^2 dx\]
41Page 58
` ∫ tan x . sec^2 x \sqrt{1 - tan^2 x} dx\ `
42Page 58
\[\int\log x\frac{\text{sin} \left\{ 1 + \left( \log x \right)^2 \right\}}{x} dx\]
43Page 58
\[\int\frac{1}{x^2} \cos^2 \left( \frac{1}{x} \right) dx\]
44Page 58
\[\int \sec^4 \text{ x tan x dx} \]
45Page 58
\[\int\frac{e^\sqrt{x} \cos \left( e^\sqrt{x} \right)}{\sqrt{x}} dx\]
46Page 58
\[\int\frac{\cos^5 x}{\sin x} dx\]
47Page 59
\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]
48Page 59
\[\int\frac{\left( x + 1 \right) e^x}{\sin^2 \left( \text{x e}^x \right)} dx\]
49Page 59
\[\int 5^{x + \tan^{- 1} x} . \left( \frac{x^2 + 2}{x^2 + 1} \right) dx\]
50Page 59
` ∫ e^{m sin ^-1 x}/ \sqrt{1-x^2} ` dx
51Page 59
\[\int\frac{\cos\sqrt{x}}{\sqrt{x}} dx\]
52Page 59
\[\int\frac{\sin \left( \tan^{- 1} x \right)}{1 + x^2} dx\]
53Page 59
\[\int\frac{\sin \left( \text{log x} \right)}{x} dx\]
54Page 59
\[\int\frac{e^{m \tan^{- 1} x}}{1 + x^2} dx\]
55Page 59
\[\int\frac{x}{\sqrt{x^2 + a^2} + \sqrt{x^2 - a^2}} dx\]
56Page 59
` ∫ x {tan^{- 1} x^2}/{1 + x^4} dx`
57Page 59
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} dx\]
58Page 59
\[\int\frac{\text{sin }\left( \text{2 + 3 log x }\right)}{x} dx\]
59Page 59
\[\ ∫ x \text{ e}^{x^2} dx\]
60Page 59
\[\int\frac{e^{2x}}{1 + e^x} dx\]
61Page 59
\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} dx\]
62Page 59
\[\int \tan^3 \text{2x sec 2x dx}\]
63Page 59
\[\int\frac{x + \sqrt{x + 1}}{x + 2} dx\]
64Page 59
\[\int 5^{5^{5^x}} 5^{5^x} 5^x dx\]
65Page 59
\[\int\frac{1}{x\sqrt{x^4 - 1}} dx\]
66Page 59
\[\int\sqrt {e^x- 1} \text{dx}\]
67Page 59
\[\int\frac{1}{\left( x + 1 \right)\left( x^2 + 2x + 2 \right)} dx\]
68Page 59
\[\int\frac{x^5}{\sqrt{1 + x^3}} dx\]
69Page 59
\[\int4 x^3 \sqrt{5 - x^2} dx\]
70Page 59
\[\int\frac{1}{\sqrt{x} + x} \text{ dx }\]
71Page 59
\[\int\frac{1}{x^2 \left( x^4 + 1 \right)^{3/4}} dx\]
72Page 59
\[\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.10 [Page 65]
1Page 65
\[\int x^2 \sqrt{x + 2} \text{ dx }\]
2Page 65
\[\int\frac{x^2}{\sqrt{x - 1}} dx\]
3Page 65
\[\int\frac{x^2}{\sqrt{3x + 4}} dx\]
4Page 65
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
5Page 65
\[\int\left( 2 x^2 + 3 \right) \sqrt{x + 2} \text{ dx }\]
6Page 65
\[\int\frac{x^2 + 3x + 1}{\left( x + 1 \right)^2} dx\]
7Page 65
\[\int\frac{x^2}{\sqrt{1 - x}} dx\]
8Page 65
\[\ \int\ x \left( 1 - x \right)^{23} dx\]
9Page 65
\[\int\frac{1}{\sqrt{x} + \sqrt[4]{x}}dx\]
10Page 65
` ∫ 1 /{x^{1/3} ( x^{1/3} -1)} ` dx
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.11 [Page 69]
3Page 69
` ∫ tan^5 x sec ^4 x dx `
6Page 69
` ∫ \sqrt{tan x} sec^4 x dx `
7Page 69
\[\int \sec^4 2x \text{ dx }\]
8Page 69
\[\int {cosec}^4 \text{ 3x } \text{ dx } \]
9Page 69
\[\int \cot^n {cosec}^2 \text{ x dx } , n \neq - 1\]
10Page 69
\[\int \cot^5 \text{ x } {cosec}^4 x\text{ dx }\]
11Page 69
\[\int \cot^5 x \text{ dx }\]
12Page 69
\[\int \cot^6 x \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.12 [Page 73]
1Page 73
\[\int \sin^4 x \cos^3 x \text{ dx }\]
2Page 73
\[\int \sin^5 x \text{ dx }\]
3Page 73
\[\int \cos^5 x \text{ dx }\]
4Page 73
\[\int \sin^5 x \cos x \text{ dx }\]
5Page 73
\[\int \sin^3 x \cos^6 x \text{ dx }\]
6Page 73
\[\int \cos^7 x \text{ dx } \]
7Page 73
\[\int x \cos^3 x^2 \sin x^2 \text{ dx }\]
8Page 73
\[\int \sin^7 x \text{ dx }\]
9Page 73
\[\int \sin^3 x \cos^5 x \text{ dx }\]
10Page 73
\[\int\frac{1}{\sin^4 x \cos^2 x} dx\]
11Page 73
\[\int\frac{1}{\sin^3 x \cos^5 x} dx\]
12Page 73
` = ∫1/{sin^3 x cos^ 2x} dx`
13Page 73
\[\int\frac{1}{\sin x \cos^3 x} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.13 [Page 79]
1Page 79
Evaluate the following integrals:
\[\int\frac{x^2}{\left( a^2 - x^2 \right)^{3/2}}dx\]
2Page 79
Evaluate the following integrals:
\[\int\frac{x^7}{\left( a^2 - x^2 \right)^5}dx\]
3Page 79
Evaluate the following integrals:
\[\int\cos\left\{ 2 \cot^{- 1} \sqrt{\frac{1 + x}{1 - x}} \right\}dx\]
4Page 79
Evaluate the following integrals:
\[\int\frac{\sqrt{1 + x^2}}{x^4}dx\]
5Page 79
Evaluate the following integrals:
\[\int\frac{1}{\left( x^2 + 2x + 10 \right)^2}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.14 [Page 83]
1Page 83
\[\int\frac{1}{a^2 - b^2 x^2} dx\]
2Page 83
` ∫ {1}/{a^2 x^2- b^2}dx`
3Page 83
\[\int\frac{1}{a^2 x^2 + b^2} dx\]
4Page 83
\[\int\frac{x^2 - 1}{x^2 + 4} dx\]
5Page 83
\[\int\frac{1}{\sqrt{1 + 4 x^2}} dx\]
6Page 83
\[\int\frac{1}{\sqrt{a^2 + b^2 x^2}} dx\]
7Page 83
\[\int\frac{1}{\sqrt{a^2 - b^2 x^2}} dx\]
8Page 83
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 + 1}} dx\]
9Page 83
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 - 1}} dx\]
10Page 83
\[\int\frac{x^4 + 1}{x^2 + 1} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.15 [Page 86]
1Page 86
\[\int\frac{1}{4 x^2 + 12x + 5} dx\]
2Page 86
\[\int\frac{1}{x^2 - 10x + 34} dx\]
3Page 86
\[\int\frac{1}{1 + x - x^2} \text{ dx }\]
4Page 86
\[\int\frac{1}{2 x^2 - x - 1} dx\]
5Page 86
\[\int\frac{1}{x^2 + 6x + 13} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.16 [Page 90]
1Page 90
\[\int\frac{\sec^2 x}{1 - \tan^2 x} dx\]
2Page 90
\[\int\frac{e^x}{1 + e^{2x}} dx\]
3Page 90
\[\int\frac{\cos x}{\sin^2 x + 4 \sin x + 5} dx\]
4Page 90
\[\int\frac{e^x}{e^{2x} + 5 e^x + 6} dx\]
5Page 90
\[\int\frac{e^{3x}}{4 e^{6x} - 9} dx\]
6Page 90
\[\int\frac{dx}{e^x + e^{- x}}\]
7Page 90
\[\int\frac{x}{x^4 + 2 x^2 + 3} dx\]
8Page 90
\[\int\frac{3 x^5}{1 + x^{12}} dx\]
9Page 90
` ∫ { x^2 dx}/{x^6 - a^6} dx `
10Page 90
\[\int\frac{x^2}{x^6 + a^6} dx\]
11Page 90
\[\int\frac{1}{x \left( x^6 + 1 \right)} dx\]
12Page 90
\[\int\frac{x}{x^4 - x^2 + 1} dx\]
13Page 90
\[\int\frac{x}{3 x^4 - 18 x^2 + 11} dx\]
14Page 90
\[\int\frac{e^x}{\left( 1 + e^x \right)\left( 2 + e^x \right)} dx\]
15Page 90
` ∫ {1} / {cos x + "cosec x" } dx `
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.17 [Page 93]
1Page 93
\[\int\frac{1}{\sqrt{2x - x^2}} dx\]
2Page 93
\[\int\frac{1}{\sqrt{8 + 3x - x^2}} dx\]
3Page 93
\[\int\frac{1}{\sqrt{5 - 4x - 2 x^2}} dx\]
4Page 93
\[\int\frac{1}{\sqrt{3 x^2 + 5x + 7}} dx\]
5Page 93
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
6Page 93
\[\int\frac{1}{\sqrt{7 - 3x - 2 x^2}} dx\]
7Page 93
\[\int\frac{1}{\sqrt{16 - 6x - x^2}} dx\]
8Page 93
\[\int\frac{1}{\sqrt{7 - 6x - x^2}} dx\]
9Page 93
\[\int\frac{1}{\sqrt{5 x^2 - 2x}} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.18 [Pages 98 - 99]
1Page 98
\[\int\frac{x}{\sqrt{x^4 + a^4}} dx\]
2Page 98
\[\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx\]
3Page 98
\[\int\frac{e^x}{\sqrt{16 - e^{2x}}} dx\]
4Page 99
\[\int\frac{\cos x}{\sqrt{4 + \sin^2 x}} dx\]
5Page 99
\[\int\frac{\sin x}{\sqrt{4 \cos^2 x - 1}} dx\]
6Page 99
\[\int\frac{x}{\sqrt{4 - x^4}} dx\]
7Page 99
\[\int\frac{1}{x\sqrt{4 - 9 \left( \log x \right)^2}} dx\]
8Page 99
\[\int\frac{\sin 8x}{\sqrt{9 + \sin^4 4x}} dx\]
9Page 99
\[\int\frac{\cos 2x}{\sqrt{\sin^2 2x + 8}} dx\]
10Page 99
\[\int\frac{\sin 2x}{\sqrt{\sin^4 x + 4 \sin^2 x - 2}} dx\]
11Page 99
\[\int\frac{\sin 2x}{\sqrt{\cos^4 x - \sin^2 x + 2}} dx\]
12Page 99
\[\int\frac{\cos x}{\sqrt{4 - \sin^2 x}} dx\]
13Page 99
\[\int\frac{1}{x^{2/3} \sqrt{x^{2/3} - 4}} dx\]
14Page 99
\[\int\frac{1}{\sqrt{\left( 1 - x^2 \right)\left\{ 9 + \left( \sin^{- 1} x \right)^2 \right\}}} dx\]
15Page 99
\[\int\frac{\cos x}{\sqrt{\sin^2 x - 2 \sin x - 3}} dx\]
16Page 99
` ∫ \sqrt{"cosec x"- 1} dx `
17Page 99
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} dx\]
18Page 99
\[\int\frac{\cos x - \sin x}{\sqrt{8 - \sin2x}}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.19 [Page 104]
1Page 104
\[\int\frac{x}{x^2 + 3x + 2} dx\]
2Page 104
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
3Page 104
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
4Page 104
\[\int\frac{2x - 3}{x^2 + 6x + 13} dx\]
5Page 104
\[\int\frac{x - 1}{3 x^2 - 4x + 3} dx\]
6Page 104
\[\int\frac{2x}{2 + x - x^2} \text{ dx }\]
7Page 104
\[\int\frac{1 - 3x}{3 x^2 + 4x + 2}\text{ dx}\]
8Page 104
\[\int\frac{2x + 5}{x^2 - x - 2} \text{ dx }\]
9Page 104
\[\int\frac{a x^3 + bx}{x^4 + c^2} dx\]
10Page 104
\[\int\frac{\left( 3 \sin x - 2 \right) \cos x}{5 - \cos^2 x - 4 \sin x} dx\]
11Page 104
\[\int\frac{x + 2}{2 x^2 + 6x + 5}\text{ dx }\]
12Page 104
Evaluate the following integrals:
\[\int\frac{5x - 2}{1 + 2x + 3 x^2} \text{ dx }\]
13Page 104
\[\int\frac{x + 5}{3 x^2 + 13x - 10}\text{ dx }\]
14Page 104
\[\int\frac{\left( 3\sin x - 2 \right)\cos x}{13 - \cos^2 x - 7\sin x}dx\]
15Page 104
\[\int\frac{x + 7}{3 x^2 + 25x + 28}\text{ dx}\]
16Page 104
\[\int\frac{x^3}{x^4 + x^2 + 1}dx\]
17Page 104
\[\int\frac{x^3 - 3x}{x^4 + 2 x^2 - 4}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.2 [Page 106]
1Page 106
\[\int\frac{x^2 + x + 1}{x^2 - x} dx\]
2Page 106
\[\int\frac{x^2 + x - 1}{x^2 + x - 6}\text{ dx }\]
3Page 106
\[\int\frac{\left( 1 - x^2 \right)}{x \left( 1 - 2x \right)} \text
{dx\]
4Page 106
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} \text{ dx }\]
5Page 106
\[\int\frac{x^2}{x^2 + 7x + 10}\text{ dx }\]
5Page 106
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
6Page 106
\[\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }\]
7Page 106
\[\int\frac{\left( x - 1 \right)^2}{x^2 + 2x + 2} dx\]
8Page 106
\[\int\frac{x^3 + x^2 + 2x + 1}{x^2 - x + 1}\text{ dx }\]
9Page 106
\[\int\frac{x^2 \left( x^4 + 4 \right)}{x^2 + 4} \text{ dx }\]
10Page 106
\[\int\frac{x^2}{x^2 + 6x + 12} \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.21 [Pages 110 - 111]
1Page 110
\[\int\frac{x}{\sqrt{x^2 + 6x + 10}} \text{ dx }\]
2Page 110
\[\int\frac{2x + 1}{\sqrt{x^2 + 2x - 1}}\text{ dx }\]
3Page 110
\[\int\frac{x + 1}{\sqrt{4 + 5x - x^2}} \text{ dx }\]
4Page 110
\[\int\frac{6x - 5}{\sqrt{3 x^2 - 5x + 1}} \text{ dx }\]
5Page 110
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
6Page 110
\[\int\frac{x}{\sqrt{8 + x - x^2}} dx\]
7Page 110
\[\int\frac{x + 2}{\sqrt{x^2 + 2x - 1}} \text{ dx }\]
8Page 110
\[\int\frac{x + 2}{\sqrt{x^2 - 1}} \text{ dx }\]
9Page 110
\[\int\frac{x - 1}{\sqrt{x^2 + 1}} \text{ dx }\]
10Page 110
\[\int\frac{x}{\sqrt{x^2 + x + 1}} \text{ dx }\]
11Page 110
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
12Page 110
\[\int\frac{2x + 5}{\sqrt{x^2 + 2x + 5}} dx\]
13Page 110
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
14Page 110
\[\int\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
15Page 111
\[\int\frac{2x + 1}{\sqrt{x^2 + 4x + 3}} \text{ dx }\]
16Page 111
\[\int\frac{2x + 3}{\sqrt{x^2 + 4x + 5}} \text{ dx }\]
17Page 111
\[\int\frac{5x + 3}{\sqrt{x^2 + 4x + 10}} \text{ dx }\]
18Page 111
Evaluate the following integrals:
\[\int\frac{x + 2}{\sqrt{x^2 + 2x + 3}} \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.22 [Page 114]
1Page 114
\[\int\frac{1}{4 \cos^2 x + 9 \sin^2 x}\text{ dx }\]
2Page 114
\[\int\frac{1}{4 \sin^2 x + 5 \cos^2 x} \text{ dx }\]
3Page 114
\[\int\frac{2}{2 + \sin 2x}\text{ dx }\]
4Page 114
\[\int\frac{\cos x}{\cos 3x} \text{ dx }\]
5Page 114
\[\int\frac{1}{1 + 3 \sin^2 x} \text{ dx }\]
6Page 114
\[\int\frac{1}{3 + 2 \cos^2 x} \text{ dx }\]
7Page 114
\[\int\frac{1}{\left( \sin x - 2 \cos x \right)\left( 2 \sin x + \cos x \right)} \text{ dx }\]
8Page 114
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
9Page 114
\[\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx\]
10Page 114
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
11Page 114
\[\int\frac{1}{\cos 2x + 3 \sin^2 x} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.23 [Page 117]
1Page 117
\[\int\frac{1}{5 + 4 \cos x} dx\]
2Page 117
\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]
3Page 117
\[\int\frac{1}{1 - 2 \sin x} \text{ dx }\]
4Page 117
\[\int\frac{1}{4 \cos x - 1} \text{ dx }\]
5Page 117
\[\int\frac{1}{1 - \sin x + \cos x} \text{ dx }\]
6Page 117
\[\int\frac{1}{3 + 2 \sin x + \cos x} \text{ dx }\]
7Page 117
\[\int\frac{1}{13 + 3 \cos x + 4 \sin x} dx\]
8Page 117
`int 1/(cos x - sin x)dx`
9Page 117
\[\int\frac{1}{\sin x + \cos x} \text{ dx }\]
10Page 117
\[\int\frac{1}{5 - 4 \cos x} \text{ dx }\]
11Page 117
\[\int\frac{1}{2 + \sin x + \cos x} \text{ dx }\]
12Page 117
\[\int\frac{1}{\sin x + \sqrt{3} \cos x} \text{ dx }\]
13Page 117
\[\int\frac{1}{\sqrt{3} \sin x + \cos x} dx\]
14Page 117
`int 1/(sin x - sqrt3 cos x) dx`
15Page 117
\[\int\frac{1}{5 + 7 \cos x + \sin x} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.24 [Page 122]
1Page 122
\[\int\frac{1}{1 - \cot x} dx\]
2Page 122
\[\int\frac{1}{1 - \tan x} \text{ dx }\]
3Page 122
\[\int\frac{3 + 2 \cos x + 4 \sin x}{2 \sin x + \cos x + 3} \text{ dx }\]
4Page 122
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
5Page 122
\[\int\frac{5 \cos x + 6}{2 \cos x + \sin x + 3} \text{ dx }\]
6Page 122
\[\int\frac{2 \sin x + 3 \cos x}{3 \sin x + 4 \cos x} dx\]
7Page 122
\[\int\frac{1}{3 + 4 \cot x} dx\]
8Page 122
\[\int\frac{2 \tan x + 3}{3 \tan x + 4} \text{ dx }\]
9Page 122
\[\int\frac{1}{4 + 3 \tan x} dx\]
10Page 122
\[\int\frac{8 \cot x + 1}{3 \cot x + 2} \text{ dx }\]
11Page 122
\[\int\frac{4 \sin x + 5 \cos x}{5 \sin x + 4 \cos x} \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.25 [Pages 133 - 134]
2Page 133
\[\int\text{ log }\left( x + 1 \right) \text{ dx }\]
3Page 133
\[\int x^3 \text{ log x dx }\]
4Page 133
\[\int x e^x \text{ dx }\]
5Page 133
\[\int x e^{2x} \text{ dx }\]
6Page 133
\[\int x^2 e^{- x} \text{ dx }\]
7Page 133
\[\int x^2 \text{ cos x dx }\]
8Page 133
\[\int x^2 \cos 2x\ \text{ dx }\]
9Page 133
\[\int x \text{ sin 2x dx }\]
10Page 133
\[\int\frac{\log \left( \log x \right)}{x} dx\]
11Page 133
\[\int x^2 \text{ cos x dx }\]
12Page 133
\[\int x\ {cosec}^2 \text{ x }\ \text{ dx }\]
14Page 133
`int"x"^"n"."log" "x" "dx"`
15Page 133
\[\int\frac{\log x}{x^n}\text{ dx }\]
16Page 133
\[\int x^2 \sin^2 x\ dx\]
19Page 133
\[\int x \sin x \cos x\ dx\]
20Page 133
` ∫ sin x log (\text{ cos x ) } dx `
21Page 133
\[\int \left( \log x \right)^2 \cdot x\ dx\]
22Page 133
\[\int e^\sqrt{x} \text{ dx }\]
23Page 133
\[\int\frac{\text{ log }\left( x + 2 \right)}{\left( x + 2 \right)^2} \text{ dx }\]
24Page 133
\[\int\frac{x + \sin x}{1 + \cos x} \text{ dx }\]
27Page 133
Evaluate the following integrals:
\[\int\frac{x \cos^{- 1} x}{\sqrt{1 - x^2}}dx\]
28Page 133
Evaluate the following integrals:
\[\int\frac{\log x}{\left( x + 1 \right)^2}dx\]
30Page 133
\[\int \sec^{- 1} \sqrt{x}\ dx\]
31Page 134
\[\int \sin^{- 1} \sqrt{x} \text{ dx }\]
33Page 134
\[\int x\left( \frac{\sec 2x - 1}{\sec 2x + 1} \right) dx\]
34Page 134
\[\int\left( x + 1 \right) \text{ e}^x \text{ log } \left( x e^x \right) dx\]
35Page 134
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx }\]
36Page 134
\[\int \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ dx }\]
37Page 134
\[\int \tan^{- 1} \left( \frac{3x - x^3}{1 - 3 x^2} \right) dx\]
38Page 134
\[\int x^2 \sin^{- 1} x\ dx\]
39Page 134
\[\int\frac{\sin^{- 1} x}{x^2} \text{ dx }\]
40Page 134
\[\int\frac{x^2 \tan^{- 1} x}{1 + x^2} \text{ dx }\]
41Page 134
\[\int \cos^{- 1} \left( 4 x^3 - 3x \right) \text{ dx }\]
42Page 134
\[\int \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) \text{ dx }\]
43Page 134
\[\int \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) \text{ dx }\]
44Page 134
\[\int\left( x + 1 \right) \text{ log x dx }\]
45Page 134
\[\int x^2 \tan^{- 1} x\text{ dx }\]
46Page 134
\[\int\left( e^\text{log x} + \sin x \right) \text{ cos x dx }\]
47Page 134
\[\int\frac{\left( x \tan^{- 1} x \right)}{\left( 1 + x^2 \right)^{3/2}} \text{ dx }\]
48Page 134
\[\int \tan^{- 1} \left( \sqrt{x} \right) \text{dx }\]
49Page 134
\[\int x^3 \tan^{- 1}\text{ x dx }\]
50Page 134
\[\int x \sin x \cos 2x\ dx\]
51Page 134
\[\int\left( \tan^{- 1} x^2 \right) x\ dx\]
52Page 134
\[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}} dx\]
53Page 134
\[\int \sin^3 \sqrt{x}\ dx\]
55Page 134
\[\int \cos^3 \sqrt{x}\ dx\]
57Page 134
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} dx\]
58Page 134
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx }\]
59Page 134
\[\int\frac{x^3 \sin^{- 1} x^2}{\sqrt{1 - x^4}} \text{ dx }\]
60Page 134
\[\int\frac{x^2 \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.26 [Page 143]
1Page 143
\[\int e^x \left( \cos x - \sin x \right) dx\]
2Page 143
\[\int e^x \left( \frac{1}{x^2} - \frac{2}{x^3} \right) dx\]
3Page 143
\[\int e^x \left( \frac{1 + \sin x}{1 + \cos x} \right) dx\]
4Page 143
\[\int e^x \left( \cot x - {cosec}^2 x \right) dx\]
5Page 143
\[\int e^x \left( \frac{x - 1}{2 x^2} \right) dx\]
6Page 143
\[\int e^x \sec x \left( 1 + \tan x \right) dx\]
7Page 143
\[\int e^x \left( \tan x - \log \cos x \right) dx\]
8Page 143
\[\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx\]
9Page 143
\[\int e^x \left( \cot x + \log \sin x \right) dx\]
10Page 143
\[\int e^x \frac{x - 1}{\left( x + 1 \right)^3} \text{ dx }\]
11Page 143
\[\int e^x \left( \frac{\sin 4x - 4}{1 - \cos 4x} \right) dx\]
12Page 14
\[\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }\]
13Page 143
\[\int e^x \frac{1 + x}{\left( 2 + x \right)^2} \text{ dx }\]
14Page 143
\[\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx }\]
15Page 143
\[\int e^x \left( \log x + \frac{1}{x} \right) dx\]
16Page 143
\[\int e^x \left( \log x + \frac{1}{x^2} \right) dx\]
17Page 143
\[\int\frac{e^x}{x}\left\{ \text{ x }\left( \log x \right)^2 + 2 \log x \right\} dx\]
18Page 143
\[\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }\]
19Page 143
∴\[\int e^{2x} \left( - \sin x + 2 \cos x \right) dx\]
20Page 143
\[\int\left( \frac{1}{\log x} - \frac{1}{\left( \log x \right)^2} \right) dx\]
21Page 143
\[\int e^x \left( \frac{\sin x \cos x - 1}{\sin^2 x} \right) dx\]
22Page 143
\[\int\left\{ \tan \left( \log x \right) + \sec^2 \left( \log x \right) \right\} dx\]
23Page 143
\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]
24Page 143
Evaluate the following integrals:
\[\int e^{2x} \left( \frac{1 - \sin2x}{1 - \cos2x} \right)dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.27 [Page 149]
1Page 149
\[\int e^{ax} \cos\ bx\ dx\]
2Page 149
\[\int e^{ax} \text{ sin} \left( bx + C \right) dx\]
3Page 149
\[\int\text{ cos }\left( \text{ log x } \right) \text{ dx }\]
4Page 149
\[\int e^{2x} \cos \left( 3x + 4 \right) \text{ dx }\]
5Page 149
\[\int e^{2x} \text{ sin x cos x dx }\]
6Page 149
\[\int e^{2x} \sin x\ dx\]
7Page 149
Evaluate the following integrals:
\[\int e^{2x} \text{ sin }\left( 3x + 1 \right) \text{ dx }\]
8Page 149
\[\int e^x \sin^2 x\ dx\]
9Page 149
\[\int\frac{1}{x^3}\text{ sin } \left( \text{ log x }\right) dx\]
10Page 149
\[\int e^{2x} \cos^2 x\ dx\]
11Page 149
\[\int e^{- 2x} \sin x\ dx\]
12Page 149
\[\int x^2 e^{x^3} \cos x^3 dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.28 [Pages 154 - 155]
1Page 154
\[\int\sqrt{3 + 2x - x^2} \text{ dx}\]
2Page 154
\[\int\sqrt{x^2 + x + 1} \text{ dx}\]
3Page 154
\[\int\sqrt{x - x^2} dx\]
4Page 154
\[\int\sqrt{1 + x - 2 x^2} \text{ dx }\]
5Page 154
\[\int\cos x \sqrt{4 - \sin^2 x}\text{ dx}\]
6Page 154
\[\int e^x \sqrt{e^{2x} + 1} \text{ dx}\]
7Page 154
\[\int\sqrt{9 - x^2}\text{ dx}\]
8Page 154
\[\int\sqrt{16 x^2 + 25} \text{ dx}\]
9Page 154
\[\int\sqrt{4 x^2 - 5}\text{ dx}\]
10Page 154
\[\int\sqrt{2 x^2 + 3x + 4} \text{ dx}\]
11Page 154
\[\int\sqrt{3 - 2x - 2 x^2} \text{ dx}\]
12Page 154
\[\int x\sqrt{x^4 + 1} \text{ dx}\]
13Page 154
\[\int x^2 \sqrt{a^6 - x^6} \text{ dx}\]
14Page 154
\[\int\frac{\sqrt{16 + \left( \log x \right)^2}}{x} \text{ dx}\]
15Page 155
\[\int\sqrt{2ax - x^2} \text{ dx}\]
16Page 155
\[\int\sqrt{3 - x^2} \text{ dx}\]
17Page 155
\[\int\sqrt{x^2 - 2x} \text{ dx}\]
18Page 155
\[\int\sqrt{2x - x^2} \text{ dx}\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.29 [Pages 158 - 159]
1Page 158
\[\int\left( x + 1 \right) \sqrt{x^2 - x + 1} \text{ dx}\]
2Page 158
\[\int\left( x + 1 \right) \sqrt{2 x^2 + 3} \text{ dx}\]
3Page 159
\[\int\left( 2x - 5 \right) \sqrt{2 + 3x - x^2} \text{ dx }\]
4Page 159
\[\int\left( x + 2 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
5Page 159
\[\int\left( 4x + 1 \right) \sqrt{x^2 - x - 2} \text{ dx }\]
6Page 159
\[\int\left( x - 2 \right) \sqrt{2 x^2 - 6x + 5} \text{ dx }\]
7Page 159
\[\int\left( x + 1 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
8Page 159
\[\int\left( 2x + 3 \right) \sqrt{x^2 + 4x + 3} \text{ dx }\]
9Page 159
\[\int\left( 2x - 5 \right) \sqrt{x^2 - 4x + 3} \text{ dx }\]
10Page 159
\[\int x\sqrt{x^2 + x} \text{ dx }\]
11Page 159
\[\int\left( x - 3 \right)\sqrt{x^2 + 3x - 18} \text{ dx }\]
12Page 159
Evaluate the following integrals:
\[\int\left( x + 3 \right)\sqrt{3 - 4x - x^2} \text{ dx }\]
13Page 159
\[\int(3x + 1) \sqrt{4 - 3x - 2 x^2} \text{ dx }\]
14Page 159
\[\int(2x + 5)\sqrt{10 - 4x - 3 x^2}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.30 [Pages 176 - 178]
1Page 176
\[\int\frac{2x + 1}{\left( x + 1 \right) \left( x - 2 \right)} dx\]
2Page 176
\[\int\frac{1}{x\left( x - 2 \right) \left( x - 4 \right)} dx\]
3Page 176
\[\int\frac{x^2 + x - 1}{x^2 + x - 6} dx\]
4Page 176
\[\int\frac{3 + 4x - x^2}{\left( x + 2 \right) \left( x - 1 \right)} dx\]
5Page 176
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
6Page 176
\[\int\frac{x^2}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
7Page 176
\[\int\frac{5x}{\left( x + 1 \right) \left( x^2 - 4 \right)} dx\]
8Page 176
\[\int\frac{x^2 + 1}{x\left( x^2 - 1 \right)} dx\]
9Page 176
\[\int\frac{2x - 3}{\left( x^2 - 1 \right) \left( 2x + 3 \right)} dx\]
10Page 176
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
11Page 176
\[\int\frac{\sin 2x}{\left( 1 + \sin x \right) \left( 2 + \sin x \right)} dx\]
12Page 176
\[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 3 \right)} dx\]
13Page 176
\[\int\frac{1}{x \log x \left( 2 + \log x \right)} dx\]
14Page 176
Evaluate the following integral :-
\[\int\frac{x^2 + x + 1}{\left( x^2 + 1 \right)\left( x + 2 \right)}dx\]
15Page 176
\[\int\frac{a x^2 + bx + c}{\left( x - a \right) \left( x - b \right) \left( x - c \right)} dx,\text{ where a, b, c are distinct}\]
16Page 176
Evaluate the following integral :-
\[\int\frac{x}{\left( x^2 + 1 \right)\left( x - 1 \right)}dx\]
17Page 176
\[\int\frac{1}{\left( x - 1 \right) \left( x + 1 \right) \left( x + 2 \right)} dx\]
18Page 176
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + 4 \right)\left( x^2 + 9 \right)}dx\]
19Page 176
\[\int\frac{5 x^2 - 1}{x \left( x - 1 \right) \left( x + 1 \right)} dx\]
20Page 176
\[\int\frac{x^2 + 6x - 8}{x^3 - 4x} dx\]
21Page 176
\[\int\frac{x^2 + 1}{\left( 2x + 1 \right) \left( x^2 - 1 \right)} dx\]
22Page 177
\[\int\frac{1}{x\left[ 6 \left( \log x \right)^2 + 7 \log x + 2 \right]} dx\]
23Page 177
\[\int\frac{1}{x\left( x^n + 1 \right)} dx\]
24Page 177
\[\int\frac{x}{\left( x^2 - a^2 \right) \left( x^2 - b^2 \right)} dx\]
25Page 177
Evaluate the following integral:
\[\int\frac{x^2 + 1}{\left( x^2 + 4 \right)\left( x^2 + 25 \right)}dx\]
26Page 177
Evaluate the following integral:
\[\int\frac{x^3 + x + 1}{x^2 - 1}dx\]
27Page 177
Evaluate the following integral:
\[\int\frac{3x - 2}{\left( x + 1 \right)^2 \left( x + 3 \right)}dx\]
28Page 177
\[\int\frac{2x + 1}{\left( x + 2 \right) \left( x - 3 \right)^2} dx\]
29Page 177
\[\int\frac{x^2 + 1}{\left( x - 2 \right)^2 \left( x + 3 \right)} dx\]
30Page 177
\[\int\frac{x}{\left( x - 1 \right)^2 \left( x + 2 \right)} dx\]
31Page 177
\[\int\frac{x^2}{\left( x - 1 \right) \left( x + 1 \right)^2} dx\]
32Page 177
\[\int\frac{x^2 + x - 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
33Page 177
\[\int\frac{2 x^2 + 7x - 3}{x^2 \left( 2x + 1 \right)} dx\]
34Page 177
\[\int\frac{5 x^2 + 20x + 6}{x^3 + 2 x^2 + x} dx\]
35Page 177
\[\int\frac{18}{\left( x + 2 \right) \left( x^2 + 4 \right)} dx\]
36Page 177
\[\int\frac{5}{\left( x^2 + 1 \right) \left( x + 2 \right)} dx\]
37Page 177
\[\int\frac{x}{\left( x + 1 \right) \left( x^2 + 1 \right)} dx\]
38Page 177
\[\int\frac{1}{1 + x + x^2 + x^3} dx\]
39Page 177
\[\int\frac{1}{\left( x + 1 \right)^2 \left( x^2 + 1 \right)} dx\]
40Page 177
\[\int\frac{2x}{x^3 - 1} dx\]
41Page 177
\[\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}\]
42Page 177
\[\int\frac{x^2}{\left( x^2 + 1 \right) \left( 3 x^2 + 4 \right)} dx\]
43Page 177
\[\int\frac{3x + 5}{x^3 - x^2 - x + 1} dx\]
44Page 177
\[\int\frac{x^3 - 1}{x^3 + x} dx\]
45Page 177
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} dx\]
46Page 177
\[\int\frac{1}{x \left( x^4 + 1 \right)} dx\]
47Page 177
Evaluate the following integral:
\[\int\frac{1}{x\left( x^3 + 8 \right)}dx\]
48Page 177
\[\int\frac{3}{\left( 1 - x \right) \left( 1 + x^2 \right)} dx\]
49Page 177
\[\int\frac{\cos x}{\left( 1 - \sin x \right)^3 \left( 2 + \sin x \right)} dx\]
50Page 177
Evaluate the following integral:
\[\int\frac{2 x^2 + 1}{x^2 \left( x^2 + 4 \right)}dx\]
51Page 177
\[\int\frac{\cos x}{\left( 1 - \sin x \right) \left( 2 - \sin x \right)} dx\]
52Page 177
\[\int\frac{2x + 1}{\left( x - 2 \right) \left( x - 3 \right)} dx\]
53Page 177
\[\int\frac{1}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)} dx\]
54Page 177
\[\int\frac{1}{x \left( x^4 - 1 \right)} dx\]
55Page 177
\[\int\frac{1}{x^4 - 1} dx\]
56Page 177
Find \[\int\frac{2x}{\left( x^2 + 1 \right) \left( x^2 + 2 \right)^2}dx\]
57Page 177
Evaluate the following integrals:
\[\int\frac{x^2}{(x - 1) ( x^2 + 1)}dx\]
58Page 177
Evaluate the following integral:
\[\int\frac{x^2}{\left( x^2 + a^2 \right)\left( x^2 + b^2 \right)}dx\]
59Page 177
\[\int\frac{1}{\cos x \left( 5 - 4 \sin x \right)} dx\]
60Page 178
\[\int\frac{1}{\sin x \left( 3 + 2 \cos x \right)} dx\]
61Page 17
\[\int\frac{1}{\sin x + \sin 2x} dx\]
62Page 178
\[\int\frac{x + 1}{x \left( 1 + x e^x \right)} dx\]
63Page 178
\[\int\frac{\left( x^2 + 1 \right) \left( x^2 + 2 \right)}{\left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
64Page 178
\[\int\frac{4 x^4 + 3}{\left( x^2 + 2 \right) \left( x^2 + 3 \right) \left( x^2 + 4 \right)} dx\]
65Page 178
\[\int\frac{x^4}{\left( x - 1 \right) \left( x^2 + 1 \right)} dx\]
66Page 178
Evaluate the following integral:
\[\int\frac{x^2}{x^4 - x^2 - 12}dx\]
67Page 178
Evaluate the following integral:
\[\int\frac{x^2}{1 - x^4}dx\]
68Page 178
Evaluate the following integral:
\[\int\frac{x^2}{x^4 + x^2 - 2}dx\]
69Page 178
\[\int\frac{( x^2 + 1) ( x^2 + 4)}{( x^2 + 3) ( x^2 - 5)} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.31 [Page 190]
1Page 190
\[\int\frac{x^2 + 1}{x^4 + x^2 + 1} \text{ dx }\]
2Page 190
\[\int\sqrt{\cot \text{θ} d } \text{ θ}\]
3Page 190
\[\int\frac{x^2 + 9}{x^4 + 81} \text{ dx }\]
4Page 190
\[\int\frac{1}{x^4 + x^2 + 1} \text{ dx }\]
5Page 190
\[\int\frac{x^2 - 3x + 1}{x^4 + x^2 + 1} \text{ dx }\]
6Page 190
\[\int\frac{x^2 + 1}{x^4 - x^2 + 1} \text{ dx }\]
7Page 190
\[\int\frac{x^2 - 1}{x^4 + 1} \text{ dx }\]
8Page 190
\[\int\frac{x^2 + 1}{x^4 + 7 x^2 + 1} 2 \text{ dx }\]
9Page 190
\[\int\frac{\left( x - 1 \right)^2}{x^4 + x^2 + 1} \text{ dx}\]
10Page 190
\[\int\frac{1}{x^4 + 3 x^2 + 1} \text{ dx }\]
11Page 190
Evaluate the following integral:
\[\int\frac{1}{\sin^4 x + \sin^2 x \cos^2 x + \cos^4 x}dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.32 [Page 196]
1Page 196
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
2Page 196
\[\int\frac{1}{\left( x - 1 \right) \sqrt{2x + 3}} \text{ dx }\]
3Page 196
\[\int\frac{x + 1}{\left( x - 1 \right) \sqrt{x + 2}} \text{ dx }\]
4Page 196
\[\int\frac{x^2}{\left( x - 1 \right) \sqrt{x + 2}}\text{ dx}\]
5Page 196
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
6Page 196
\[\int\frac{1}{\left( x^2 + 1 \right) \sqrt{x}} \text{ dx }\]
7Page 196
\[\int\frac{x}{\left( x^2 + 2x + 2 \right) \sqrt{x + 1}} \text{ dx}\]
8Page 196
\[\int\frac{1}{\left( x - 1 \right) \sqrt{x^2 + 1}} \text{ dx }\]
9Page 196
\[\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }\]
10Page 196
\[\int\frac{1}{\left( x^2 - 1 \right) \sqrt{x^2 + 1}} \text{ dx }\]
11Page 176
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 1}} \text{ dx }\]
12Page 196
\[\int\frac{1}{\left( 1 + x^2 \right) \sqrt{1 - x^2}} \text{ dx }\]
13Page 176
\[\int\frac{1}{\left( 2 x^2 + 3 \right) \sqrt{x^2 - 4}} \text{ dx }\]
14Page 196
\[\int\frac{x}{\left( x^2 + 4 \right) \sqrt{x^2 + 9}} \text{ dx}\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Very Short Answers [Pages 197 - 198]
1Page 197
Write a value of
\[\int\frac{1 + \cot x}{x + \log \sin x} \text{ dx }\]
2Page 197
Write a value of
\[\int e^{3 \text{ log x}} x^4\text{ dx}\]
3Page 197
Write a value of
\[\int x^2 \sin x^3 \text{ dx }\]
4Page 197
Write a value of
\[\int \tan^3 x \sec^2 x \text{ dx }\].
5Page 197
Write a value of
\[\int e^x \left( \sin x + \cos x \right) \text{ dx}\]
6Page 197
Write a value of
\[\int \tan^6 x \sec^2 x \text{ dx }\] .
7Page 197
Write a value of
\[\int\frac{\cos x}{3 + 2 \sin x}\text{ dx}\]
8Page 197
Write a value of
\[\int e^x \sec x \left( 1 + \tan x \right) \text{ dx }\]
9Page 197
Write a value of
\[\int\frac{\log x^n}{x} \text{ dx}\]
10Page 197
Write a value of
\[\int\frac{\left( \log x \right)^n}{x} \text{ dx }\]
11Page 197
Write a value of
\[\int e^{\text{ log sin x }}\text{ cos x}. \text{ dx}\]
12Page 197
Write a valoue of \[\int \sin^3 x \cos x\ dx\]
13Page 197
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
14Page 197
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
15Page 179
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
16Page 197
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
17Page 197
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
18Page 197
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
19Page 197
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
20Page 197
Write a value of\[\int \log_e x\ dx\].
21Page 197
Write a value of\[\int a^x e^x \text{ dx }\]
22Page 197
Write a value of
\[\int e^{2 x^2 + \ln x} \text{ dx}\]
23Page 197
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
24Page 197
Write a value of\[\int\frac{\cos x}{\sin x \log \sin x} dx\]
25Page 197
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
26Page 197
Write a value of
\[\int\frac{a^x}{3 + a^x} \text{ dx}\]
27Page 197
Write a value of
\[\int\frac{1 + \log x}{3 + x \log x} \text{ dx }\] .
28Page 197
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
29Page 197
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
30Page 197
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
31Page 197
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
32Page 197
Write a value of\[\int e^{ax} \cos\ bx\ dx\].
33Page 198
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
34Page 198
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
35Page 198
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
36Page 198
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
37Page 198
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
38Page 198
Evaluate:\[\int\frac{x^2}{1 + x^3} \text{ dx }\] .
39Page 198
Evaluate:
\[\int\frac{x^2 + 4x}{x^3 + 6 x^2 + 5} \text{ dx }\]
40Page 198
Evaluate:\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} \text{ dx }\]
41Page 198
Evaluate:\[\int\frac{\sin \sqrt{x}}{\sqrt{x}} \text{ dx }\]
42Page 198
Evaluate:\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} \text{ dx }\]
43Page 198
Evaluate:\[\int\frac{\left( 1 + \log x \right)^2}{x} \text{ dx }\]
44Page 198
Evaluate:\[\int \sec^2 \left( 7 - 4x \right) \text{ dx }\]
45Page 198
Evaluate:\[\int\frac{\log x}{x} \text{ dx }\]
46Page 198
Evaluate: \[\int 2^x \text{ dx }\]
47Page 198
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
48Page 198
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
49Page 198
Evaluate: \[\int\frac{x^3 - x^2 + x - 1}{x - 1} \text{ dx }\]
50Page 198
Evaluate:\[\int\frac{e\tan^{- 1} x}{1 + x^2} \text{ dx }\]
51Page 198
Evaluate: \[\int\frac{1}{\sqrt{1 - x^2}} \text{ dx }\]
52Page 198
Write the value of\[\int\sec x \left( \sec x + \tan x \right)\text{ dx }\]
53Page 198
Evaluate: \[\int\frac{1}{x^2 + 16}\text{ dx }\]
54Page 198
Evaluate: \[\int\left( 1 - x \right)\sqrt{x}\text{ dx }\]
55Page 198
Evaluate: \[\int\frac{x + \cos6x}{3 x^2 + \sin6x}\text{ dx }\]
56Page 198
\[\text{ If } \int\left( \frac{x - 1}{x^2} \right) e^x dx = f\left( x \right) e^x + C, \text{ then write the value of f}\left( x \right) .\]
57Page 198
\[If \int e^x \left( \tan x + 1 \right)\text{ sec x dx } = e^x f\left( x \right) + C, \text{ then write the value of f}\left( x \right) .\]
58Page 198
Evaluate: \[\int\frac{2}{1 - \cos2x}\text{ dx }\]
59Page 198
Write the anti-derivative of \[\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .\]
60Page 198
Evaluate:
\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]
61Page 198
Evaluate:
`∫ (1)/(sin^2 x cos^2 x) dx`
62Page 198
Evaluate : \[\int\frac{1}{x(1 + \log x)} \text{ dx}\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals MCQ [Pages 199 - 203]
1Page 199
\[\int\frac{x}{4 + x^4} \text{ dx }\] is equal to
\[\frac{1}{4} \tan^{- 1} x^2 + C\]
\[\frac{1}{4} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
\[\frac{1}{2} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
none of these
2Page 199
\[\int\frac{1}{\cos x + \sqrt{3} \sin x} \text{ dx } \] is equal to
` log tan (x/3 + π / 2) + C `
\[\text{ log tan} \left( \frac{x}{2} - \frac{\pi}{3} \right) + C\]
` 1/2 log tan (x/2 + π /3 ) + C `
none of these
3Page 200
` \int \text{ x} \text{ sec x}^2 \text{ dx is equal to }`
\[\frac{1}{2}\] log (sec x2 + tan x2) + C
\[\frac{x^2}{2}\] log (sec x2 + tan x2) + C
2 log (sec x2 + tan x2) + C
none of these
4Page 200
If \[\int\frac{1}{5 + 4 \sin x} dx = A \tan^{- 1} \left( B \tan\frac{x}{2} + \frac{4}{3} \right) + C,\] then
A =\[\frac{2}{3}\], B =\[\frac{5}{3}\]
A =\[\frac{1}{3}\], B = \[\frac{2}{3}\]
A =\[- \frac{2}{3}\], B =\[\frac{5}{3}\]
A =\[\frac{1}{3}\], B =\[- \frac{5}{3}\]
5Page 200
\[\int x^{\sin x} \left( \frac{\sin x}{x} + \cos x . \log x \right) dx\] is equal to
6Page 200
Integration of \[\frac{1}{1 + \left( \log_e x \right)^2}\] with respect to loge x is
\[\frac{\tan^{- 1} \left( \log_e x \right)}{x} + C\]
\[\tan^{- 1} \left( \log_e x \right) + C\]
\[\frac{\tan^{- 1} x}{x} + C\]
none of these
7Page 200
If \[\int\frac{\cos 8x + 1}{\tan 2x - \cot 2x} dx\]
\[- \frac{1}{16}\]
\[\frac{1}{8}\]
\[\frac{1}{16}\]
\[- \frac{1}{8}\]
8Page 200
If \[\int\frac{\sin^8 x - \cos^8 x}{1 - 2 \sin^2 x \cos^2 x} dx\]
9Page 200
\[\int\left( x - 1 \right) e^{- x} dx\] is equal to
− xex + C
xex + C
− xe−x + C
xe−x + C
10Page 200
If `int(2x^(1/2))/(x^2) dx = k . 2^(1/x) + C`, then k is equal to ______.
`(-1)/(log 2)`
− log 2
−1
`1/2`
11Page 200
\[\int\frac{1}{1 + \tan x} dx =\]
12Page 200
\[\int \left| x \right|^3 dx\] is equal to
13Page 200
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
14Page 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
15Page 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
16Page 201
\[\int\frac{1}{7 + 5 \cos x} dx =\]
17Page 201
\[\int\frac{1}{1 - \cos x - \sin x} dx =\]
\[\log\left| 1 + \cot\frac{x}{2} \right| + C\]
\[\log\left| 1 - \tan\frac{x}{2} \right| + C\]
\[\log\left| 1 - \cot\frac{x}{2} \right| + C\]
\[\log\left| 1 + \tan\frac{x}{2} \right| + C\]
18Page 201
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
\[\frac{e^x}{x + 4} + C\]
\[\frac{e^x}{x + 3} + C\]
\[\frac{1}{\left( x + 4 \right)^2} + C\]
\[\frac{e^x}{\left( x + 4 \right)^2} + C\]
19Page 201
\[\int\frac{\sin x}{3 + 4 \cos^2 x} dx\]
log (3 + 4 cos2 x) + C
\[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{\cos x}{\sqrt{3}} \right) + C\]
\[- \frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
\[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
20Page 201
\[\int e^x \left( \frac{1 - \sin x}{1 - \cos x} \right) dx\]
\[- e^x \tan\frac{x}{2} + C\]
\[- e^x \cot\frac{x}{2} + C\]
\[- \frac{1}{2} e^x \tan\frac{x}{2} + C\]
\[- \frac{1}{2} e^x \cot\frac{x}{2} + C\]
21Page 201
\[\int\frac{2}{\left( e^x + e^{- x} \right)^2} dx\]
\[\frac{- e^{- x}}{e^x + e^{- x}} + C\]
\[- \frac{1}{e^x + e^{- x}} + C\]
\[\frac{- 1}{\left( e^x + 1 \right)^2} + C\]
\[\frac{1}{e^x - e^{- x}} + C\]
22Page 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
23Page 201
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\frac{1}{3} \tan^2 x + C\]
\[\frac{1}{2} \tan^2 x + C\]
\[\frac{1}{3} \tan^3 x + C\]
24Page 202
The primitive of the function \[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}\]
\[\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
\[\log_e a \cdot a^{x + \frac{1}{x}}\]
\[\frac{a^{x + \frac{1}{x}}}{x} \log_e a\]
\[x\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
25Page 202
The value of \[\int\frac{1}{x + x \log x} dx\] is
1 + log x
x + log x
x log (1 + log x)
log (1 + log x)
26Page 202
\[\int\sqrt{\frac{x}{1 - x}} dx\] is equal to
\[\sin^{- 1} \sqrt{x} + C\]
\[\sin^{- 1} \left\{ \sqrt{x} - \sqrt{x \left( 1 - x \right)} \right\} + C\]
\[\sin^{- 1} \left\{ \sqrt{x \left( 1 - x \right)} \right\} + C\]
\[\sin^{- 1} \sqrt{x} - \sqrt{x \left( 1 - x \right)} + C\]
27Page 202
\[\int e^x \left\{ f\left( x \right) + f'\left( x \right) \right\} dx =\]
ex f (x) + C
ex + f (x)
2ex f (x)
ex − f (x)
28Page 202
The value of \[\int\frac{\sin x + \cos x}{\sqrt{1 - \sin 2x}} dx\] is equal to
30Page 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
31Page 202
\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\] is equal to
32Page 202
\[\int\frac{x^9}{\left( 4 x^2 + 1 \right)^6}dx\] is equal to
\[ \frac{1}{5x} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C\]
\[ \frac{1}{5} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C\]
\[ \frac{1}{10x} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C\]
\[ \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C\]
33Page 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
\[ a = \frac{1}{3}, b = 1\]
\[a = - \frac{1}{3}, b = 1\]
\[ a = - \frac{1}{3}, b = - 1\]
\[ a = \frac{1}{3}, b = - 1\]
34Page 202
\[\int\frac{x^3}{x + 1}dx\] is equal to
\[ x + \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 - x \right| + C\]
\[ x + \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 - x \right| + C\]
\[ x - \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 + x \right| + C\]
\[ x - \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 + x \right| + C\]
35Page 203
If \[\int\frac{1}{\left( x + 2 \right)\left( x^2 + 1 \right)}dx = a\log\left| 1 + x^2 \right| + b \tan^{- 1} x + \frac{1}{5}\log\left| x + 2 \right| + C,\] then
\[ a = - \frac{1}{10}, b = - \frac{2}{5}\]
\[a = \frac{1}{10}, b = - \frac{2}{5}\]
\[ a = - \frac{1}{10}, b = \frac{2}{5}\]
\[ a = \frac{1}{10}, b = \frac{2}{5}\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Revision Excercise [Pages 203 - 205]
1Page 203
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
2Page 203
\[\int\frac{1 - x^4}{1 - x} \text{ dx }\]
3Page 203
\[\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }\]
4Page 203
\[\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }\]
5Page 203
\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]
6Page 203
\[\int\frac{\left( 2^x + 3^x \right)^2}{6^x} \text{ dx }\]
7Page 203
\[\int\frac{\sin x}{1 + \sin x} \text{ dx }\]
8Page 203
\[\int\frac{x^4 + x^2 - 1}{x^2 + 1} \text{ dx}\]
9Page 203
\[\int \sec^2 x \cos^2 2x \text{ dx }\]
10Page 203
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
13Page 203
\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}\]
14Page 203
\[\int\frac{1}{\left( \sin^{- 1} x \right) \sqrt{1 - x^2}} \text{ dx} \]
15Page 203
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
16Page 203
\[\int\frac{1}{e^x + 1} \text{ dx }\]
17Page 203
\[\int\frac{e^x - 1}{e^x + 1} \text{ dx}\]
18Page 203
\[\int\frac{1}{e^x + e^{- x}} dx\]
19Page 203
\[\int\frac{\cos^7 x}{\sin x} dx\]
20Page 203
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
21Page 203
\[\int\text{ cos x cos 2x cos 3x dx}\]
22Page 203
\[\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \text{ dx}\]
23Page 203
\[\int\frac{\sin x - \cos x}{\sqrt{\sin 2x}} \text{ dx }\]
24Page 203
\[\int\frac{1}{\text{ sin} \left( x - a \right) \text{ sin } \left( x - b \right)} \text{ dx }\]
25Page 203
\[\int\frac{1}{\text{ cos }\left( x - a \right) \text{ cos }\left( x - b \right)} \text{ dx }\]
26Page 203
\[\int\frac{\sin x}{\sqrt{1 + \sin x}} dx\]
27Page 203
\[\int\frac{\sin x}{\cos 2x} \text{ dx }\]
33Page 203
\[\int\frac{x^2}{\left( x - 1 \right)^3} dx\]
34Page 203
\[\int x\sqrt{2x + 3} \text{ dx }\]
35Page 203
\[\int\frac{x^3}{\left( 1 + x^2 \right)^2} \text{ dx }\]
36Page 203
\[\int x \sin^5 x^2 \cos x^2 dx\]
37Page 203
\[\int \sin^3 x \cos^4 x\ \text{ dx }\]
40Page 203
\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]
41Page 203
\[\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }\]
42Page 203
\[\int\frac{1}{\sqrt{x^2 - a^2}} \text{ dx }\]
43Page 203
\[\int\frac{1}{\sqrt{x^2 + a^2}} \text{ dx }\]
44Page 203
\[\int\frac{1}{4 x^2 + 4x + 5} dx\]
45Page 203
\[\int\frac{1}{x^2 + 4x - 5} \text{ dx }\]
46Page 203
\[\int\frac{1}{1 - x - 4 x^2}\text{ dx }\]
47Page 203
\[\int\frac{1}{3 x^2 + 13x - 10} \text{ dx }\]
48Page 203
\[\int\frac{\sin x}{\sqrt{\cos^2 x - 2 \cos x - 3}} \text{ dx }\]
49Page 204
\[\int\sqrt{\text{ cosec x} - 1} \text{ dx }\]
50Page 204
\[\int\frac{1}{\sqrt{3 - 2x - x^2}} \text{ dx}\]
51Page 204
\[\int\frac{x + 1}{x^2 + 4x + 5} \text{ dx}\]
52Page 204
\[\int\frac{5x + 7}{\sqrt{\left( x - 5 \right) \left( x - 4 \right)}} \text{ dx }\]
53Page 204
\[\int\sqrt{\frac{1 + x}{x}} \text{ dx }\]
54Page 204
\[\int\sqrt{\frac{1 - x}{x}} \text{ dx}\]
55Page 204
\[\int\frac{\sqrt{a} - \sqrt{x}}{1 - \sqrt{ax}}\text{ dx }\]
56Page 204
\[\int\frac{1}{\left( \sin x - 2 \cos x \right) \left( 2 \sin x + \cos x \right)} \text{ dx }\]
57Page 204
\[\int\frac{1}{4 \sin^2 x + 4 \sin x \cos x + 5 \cos^2 x} \text{ dx }\]
58Page 204
\[\int\frac{1}{a + b \tan x} \text{ dx }\]
59Page 204
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
60Page 204
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
61Page 204
\[\int\frac{x^3}{\sqrt{x^8 + 4}} \text{ dx }\]
62Page 204
\[\int\frac{1}{2 - 3 \cos 2x} \text{ dx }\]
63Page 204
\[\int\frac{\cos x}{\frac{1}{4} - \cos^2 x} \text{ dx }\]
64Page 204
\[\int\frac{1}{1 + 2 \cos x} \text{ dx }\]
65Page 204
\[\int\frac{1}{1 - 2 \sin x} \text{ dx }\]
66Page 204
\[\int\frac{1}{\sin x \left( 2 + 3 \cos x \right)} \text{ dx }\]
67Page 204
\[\int\frac{1}{\sin x + \sin 2x} \text{ dx }\]
68Page 204
\[\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}\]
69Page 204
\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]
71Page 204
\[\int {cosec}^4 2x\ dx\]
72Page 204
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
73Page 204
\[\int\frac{1}{2 + \cos x} \text{ dx }\]
74Page 204
\[\int\sqrt{\frac{a + x}{x}}dx\]
75Page 204
\[\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} \text{ dx}\]
76Page 204
\[\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }\]
77Page 204
\[\int\frac{\cos^5 x}{\sin x} \text{ dx }\]
78Page 204
\[\int\frac{\sin^6 x}{\cos x} \text{ dx }\]
79Page 204
\[\int\frac{\sin^2 x}{\cos^6 x} \text{ dx }\]
81Page 204
\[\int \tan^5 x\ \sec^3 x\ dx\]
82Page 204
\[\int \tan^3 x\ \sec^4 x\ dx\]
83Page 204
\[\int\frac{1}{\sec x + cosec x}\text{ dx }\]
84Page 204
\[\int\sqrt{a^2 + x^2} \text{ dx }\]
85Page 204
\[\int\sqrt{x^2 - a^2} \text{ dx}\]
86Page 204
\[\int\sqrt{a^2 - x^2}\text{ dx }\]
87Page 204
\[\int\sqrt{3 x^2 + 4x + 1}\text{ dx }\]
88Page 204
\[\int\sqrt{1 + 2x - 3 x^2}\text{ dx } \]
89Page 204
\[\int x\sqrt{1 + x - x^2}\text{ dx }\]
90Page 204
\[\int\left( 2x + 3 \right) \sqrt{4 x^2 + 5x + 6} \text{ dx}\]
91Page 204
\[ \int\left( 1 + x^2 \right) \ \cos 2x \ dx\]
93Page 204
\[\int\frac{\log \left( \log x \right)}{x} \text{ dx}\]
96Page 204
\[\int \left( x + 1 \right)^2 e^x \text{ dx }\]
97Page 204
\[\int\log \left( x + \sqrt{x^2 + a^2} \right) \text{ dx}\]
98Page 204
\[\int\frac{\log x}{x^3} \text{ dx }\]
99Page 204
\[\int\frac{\log \left( 1 - x \right)}{x^2} \text{ dx}\]
100Page 204
\[\int x^3 \left( \log x \right)^2\text{ dx }\]
101Page 204
\[\int\frac{1}{x \sqrt{1 + x^n}} \text{ dx}\]
102Page 204
\[\int\frac{x^2}{\sqrt{1 - x}} \text{ dx }\]
103Page 204
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
104Page 204
\[\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }\]
105Page 204
\[\int x\sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
106Page 204
\[\int\frac{1}{x\sqrt{1 + x^3}} \text{ dx}\]
107Page 204
\[\int\frac{\sin x + \cos x}{\sin^4 x + \cos^4 x} \text{ dx }\]
108Page 204
\[\int x^2 \tan^{- 1} x\ dx\]
109Page 205
\[\int \tan^{- 1} \sqrt{x}\ dx\]
110Page 205
\[\int \sin^{- 1} \sqrt{x}\ dx\]
111Page 204
\[\int \sec^{- 1} \sqrt{x}\ dx\]
112Page 204
\[\int \tan^{- 1} \sqrt{\frac{1 - x}{1 + x}} \text{ dx }\]
113Page 205
\[\int \sin^{- 1} \sqrt{\frac{x}{a + x}} \text{ dx}\]
114Page 205
\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx}\]
115Page 205
\[\int \left( \sin^{- 1} x \right)^3 dx\]
116Page 204
\[\int \cos^{- 1} \left( 1 - 2 x^2 \right) \text{ dx }\]
117Page 205
\[\int\frac{x \sin^{- 1} x}{\left( 1 - x^2 \right)^{3/2}} \text{ dx}\]
118Page 205
\[\int e^{2x} \left( \frac{1 + \sin 2x}{1 + \cos 2x} \right) dx\]
119Page 205
\[\int\frac{\sqrt{1 - \sin x}}{1 + \cos x} e^{- x/2} \text{ dx}\]
120Page 205
\[\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }\]
121Page 205
\[\int\frac{e^{m \tan^{- 1} x}}{\left( 1 + x^2 \right)^{3/2}} \text{ dx}\]
122Page 205
\[\int\frac{x^2}{\left( x - 1 \right)^3 \left( x + 1 \right)} \text{ dx}\]
123Page 205
\[\int\frac{x}{x^3 - 1} \text{ dx}\]
124Page 205
\[\int\frac{1}{1 + x + x^2 + x^3} \text{ dx }\]
125Page 205
\[\int\frac{1}{\left( x^2 + 2 \right) \left( x^2 + 5 \right)} \text{ dx}\]
126Page 205
\[\int\frac{x^2 - 2}{x^5 - x} \text{ dx}\]
127Page 205
\[\int\sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \text{ dx}\]
128Page 205
\[\int\frac{x^2 + x + 1}{\left( x + 1 \right)^2 \left( x + 2 \right)} \text{ dx}\]
129Page 205
\[\int\frac{\sin 4x - 2}{1 - \cos 4x} e^{2x} \text{ dx}\]
130Page 205
\[\int\frac{\cot x + \cot^3 x}{1 + \cot^3 x} \text{ dx}\]
Solutions for 19: Indefinite Integrals
Exercise 19.01Exercise 19.02Exercise 19.03Exercise 19.04Exercise 19.05Exercise 19.06Exercise 19.07Exercise 19.08Exercise 19.09Exercise 19.10Exercise 19.11Exercise 19.12Exercise 19.13Exercise 19.14Exercise 19.15Exercise 19.16Exercise 19.17Exercise 19.18Exercise 19.19Exercise 19.2Exercise 19.21Exercise 19.22Exercise 19.23Exercise 19.24Exercise 19.25Exercise 19.26Exercise 19.27Exercise 19.28Exercise 19.29Exercise 19.30Exercise 19.31Exercise 19.32Very Short AnswersMCQRevision Excercise
RD Sharma solutions for Mathematics [English] Class 12 chapter 19 - Indefinite Integrals
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Concepts covered in Mathematics [English] Class 12 chapter 19 Indefinite Integrals are Introduction of Integrals, Integration as an Inverse Process of Differentiation, Properties of Indefinite Integral, Methods of Integration> Integration by Substitution, Methods of Integration>Integration Using Trigonometric Identities, Integrals of Some Particular Functions, Methods of Integration> Integration Using Partial Fraction, Methods of Integration> Integration by Parts, Fundamental Theorem of Integral Calculus, Definite Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Overview of Integrals.
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