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Chapters
2: Functions
3: Binary Operations
4: Inverse Trigonometric Functions
5: Algebra of Matrices
6: Determinants
7: Adjoint and Inverse of a Matrix
8: Solution of Simultaneous Linear Equations
9: Continuity
10: Differentiability
11: Differentiation
12: Higher Order Derivatives
13: Derivative as a Rate Measurer
14: Differentials, Errors and Approximations
15: Mean Value Theorems
16: Tangents and Normals
17: Increasing and Decreasing Functions
18: Maxima and Minima
▶ 19: Indefinite Integrals
20: Definite Integrals
21: Areas of Bounded Regions
22: Differential Equations
23: Algebra of Vectors
24: Scalar Or Dot Product
25: Vector or Cross Product
26: Scalar Triple Product
27: Direction Cosines and Direction Ratios
28: Straight Line in Space
29: The Plane
30: Linear programming
31: Probability
32: Mean and Variance of a Random Variable
33: Binomial Distribution
![RD Sharma solutions for Mathematics [English] Class 12 chapter 19 - Indefinite Integrals - Shaalaa.com](/images/9788193663011-mathematics-english-class-12_6:be05c27f33094688837f0fdb2cb69ac3.jpg "RD Sharma solutions for Mathematics [English] Class 12 chapter 19 - Indefinite Integrals")
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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.
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.01 [Page 4]
Evaluate of the following integral:
(i) \[\int x^4 dx\]
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate of the following integral:
Evaluate:
Evaluate:
Evaluate :
Evaluate:
Evaluate:
Evaluate:
Evaluate:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.02 [Pages 14 - 15]
\[\int\left\{ x^2 + e^{\log x}+ \left( \frac{e}{2} \right)^x \right\} dx\]
\[\int\sqrt{x}\left( 3 - 5x \right) dx\]
\[\int\frac{5 x^4 + 12 x^3 + 7 x^2}{x^2 + x} dx\]
` ∫ {cosec x} / {"cosec x "- cot x} ` dx
If f' (x) = x − \[\frac{1}{x^2}\] and f (1) \[\frac{1}{2}, find f(x)\]
If f' (x) = x + b, f(1) = 5, f(2) = 13, find f(x)
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
If f' (x) = a sin x + b cos x and f' (0) = 4, f(0) = 3, f
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.03 [Pages 23 - 24]
` ∫ 1/ {1+ cos 3x} ` dx
\[\int \left( e^x + 1 \right)^2 e^x dx\]
\[\int \tan^2 \left( 2x - 3 \right) dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.04 [Page 30]
\[\int\frac{x^2 + 5x + 2}{x + 2} dx\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.05 [Page 33]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.06 [Page 36]
\[\int \sin^3 \left( 2x + 1 \right) \text{dx}\]
`∫ cos ^4 2x dx `
` ∫ sin x \sqrt (1-cos 2x) dx `
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.07 [Page 38]
Integrate the following integrals:
Integrate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.08 [Pages 47 - 48]
Evaluate the following integrals:
` ∫ {sec x "cosec " x}/{log ( tan x) }` dx
` ∫ {sin 2x} /{a cos^2 x + b sin^2 x } ` dx
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.09 [Pages 57 - 59]
` = ∫ root (3){ cos^2 x} sin x dx `
` ∫ cot^3 x "cosec"^2 x dx `
\[\int\frac{\left\{ e^{\sin^{- 1} }x \right\}^2}{\sqrt{1 - x^2}} dx\]
\[\int\frac{1}{\sqrt{1 - x^2} \left( \sin^{- 1} x \right)^2} dx\]
\[\int\frac{\cot x}{\sqrt{\sin x}} dx\]
\[\int\frac{\sqrt{\tan x}}{\sin x \cos x} dx\]
\[\int\frac{1}{x} \left( \log x \right)^2 dx\]
` ∫ e^{m sin ^-1 x}/ \sqrt{1-x^2} ` dx
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.10 [Page 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]
` ∫ tan x sec^4 x dx `
` ∫ tan^5 x dx `
` ∫ \sqrt{tan x} sec^4 x dx `
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.12 [Page 73]
` = ∫1/{sin^3 x cos^ 2x} dx`
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.13 [Page 79]
Evaluate the following integrals:
Evaluate the following integrals:
Evaluate the following integrals:
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.14 [Page 83]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.15 [Page 86]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.16 [Page 90]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.17 [Page 93]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.18 [Pages 98 - 99]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.19 [Page 104]
` ∫ {x-3} /{ x^2 + 2x - 4 } dx `
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.2 [Page 106]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.21 [Pages 110 - 111]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
\[\int\frac{x}{\sqrt{8 + x - x^2}} dx\]
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.22 [Page 114]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.23 [Page 117]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.24 [Page 122]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.25 [Pages 133 - 134]
\[\int x\ {cosec}^2 \text{ x }\ \text{ dx }\]
Evaluate the following integrals:
Evaluate the following integrals:
\[\int\left( e^\text{log x} + \sin x \right) \text{ cos x dx }\]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.26 [Page 143]
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.27 [Page 149]
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.28 [Pages 154 - 155]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.29 [Pages 158 - 159]
Evaluate the following integrals:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.30 [Pages 176 - 178]
Evaluate the following integral :-
Evaluate the following integral :-
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integrals:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
Evaluate the following integral:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.31 [Page 190]
Evaluate the following integral:
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Exercise 19.32 [Page 196]
RD Sharma solutions for Mathematics [English] Class 12 19 Indefinite Integrals Very Short Answers [Pages 197 - 198]
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Write a value of
Write a value of\[\int\frac{\sin x}{\cos^3 x} \text{ dx }\]
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
Evaluate:\[\int\frac{x^2}{1 + x^3} \text{ dx }\] .
Evaluate:
Evaluate:\[\int\frac{\sec^2 \sqrt{x}}{\sqrt{x}} \text{ dx }\]
Evaluate:\[\int\frac{\sin \sqrt{x}}{\sqrt{x}} \text{ dx }\]
Evaluate:\[\int\frac{\cos \sqrt{x}}{\sqrt{x}} \text{ dx }\]
Evaluate:\[\int\frac{\left( 1 + \log x \right)^2}{x} \text{ dx }\]
Evaluate:\[\int \sec^2 \left( 7 - 4x \right) \text{ dx }\]
Evaluate:\[\int\frac{\log x}{x} \text{ dx }\]
Evaluate: \[\int 2^x \text{ dx }\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate: \[\int\frac{x^3 - x^2 + x - 1}{x - 1} \text{ dx }\]
Evaluate:\[\int\frac{e\tan^{- 1} x}{1 + x^2} \text{ dx }\]
Evaluate: \[\int\frac{1}{\sqrt{1 - x^2}} \text{ dx }\]
Write the value of\[\int\sec x \left( \sec x + \tan x \right)\text{ dx }\]
Evaluate: \[\int\frac{1}{x^2 + 16}\text{ dx }\]
Evaluate: \[\int\left( 1 - x \right)\sqrt{x}\text{ dx }\]
Evaluate: \[\int\frac{x + \cos6x}{3 x^2 + \sin6x}\text{ dx }\]
Evaluate: \[\int\frac{2}{1 - \cos2x}\text{ dx }\]
Write the anti-derivative of \[\left( 3\sqrt{x} + \frac{1}{\sqrt{x}} \right) .\]
Evaluate:
\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]
Evaluate:
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]
\[\frac{1}{4} \tan^{- 1} x^2 + C\]
\[\frac{1}{4} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
\[\frac{1}{2} \tan^{- 1} \left( \frac{x^2}{2} \right)\]
none of these
` 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
\[\frac{1}{2}\] 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
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 =\[- \frac{5}{3}\]
xsin x + C
xsin x cos x + C
\[\frac{\left( x^{\sin x} \right)^2}{2} + C\]
none of these
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
If \[\int\frac{\cos 8x + 1}{\tan 2x - \cot 2x} dx\]
\[- \frac{1}{16}\]
\[\frac{1}{8}\]
\[\frac{1}{16}\]
\[- \frac{1}{8}\]
If \[\int\frac{\sin^8 x - \cos^8 x}{1 - 2 \sin^2 x \cos^2 x} dx\]
-1/2
1/2
-1
1
− xex + C
xex + C
− xe−x + C
xe−x + C
If \[\int\frac{2^{1/x}}{x^2} dx = k 2^{1/x} + C,\] then k is equal to
\[- \frac{1}{\log_e 2}\]
− loge 2
`-1`
\[\frac{1}{2}\]
loge (x + sin x) + C
loge (sin x + cos x) + C
\[2 \sec^2 \frac{x}{2} + C\]
\[\frac{1}{2}\] [x + log (sin x + cos x)] + C
\[\frac{- x^4}{4} + C\]
\[\frac{\left| x \right|^4}{4} + C\]
\[\frac{x^4}{4} + C\]
none of these
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
2 cos \[\sqrt{x}\]
\[\sqrt{\frac{\cos x}{x}} + C\]
sin \[\sqrt{x} + C\]
2 sin \[\sqrt{x} + C\]
ex cot x + C
−ex cot x + C
ex cosec x + C
−ex cosec x + C
tan 7x + C
- \[\frac{\tan^7 x}{7} + C\]
- \[\frac{\tan 7x}{7} + C\]
sec7 x + C
- \[\frac{1}{\sqrt{6}} \tan^{- 1} \left( \frac{1}{\sqrt{6}}\tan\frac{x}{2} \right) + C\]
\[\frac{1}{\sqrt{3}} \tan^{- 1} \left( \frac{1}{\sqrt{3}}\tan\frac{x}{2} \right) + C\]
- \[\frac{1}{4} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C\]
- \[\frac{1}{7} \tan^{- 1} \left( \tan\frac{x}{2} \right) + C\]
- \[\log\left| 1 + \cot\frac{x}{2} \right| + C\]
- \[\log\left| 1 - \tan\frac{x}{2} \right| + C\]
- \[\log\left| 1 - \cot\frac{x}{2} \right| + C\]
- \[\log\left| 1 + \tan\frac{x}{2} \right| + C\]
\[\int\frac{x + 3}{\left( x + 4 \right)^2} e^x dx =\]
\[\frac{e^x}{x + 4} + C\]
\[\frac{e^x}{x + 3} + C\]
\[\frac{1}{\left( x + 4 \right)^2} + C\]
\[\frac{e^x}{\left( x + 4 \right)^2} + C\]
log (3 + 4 cos2 x) + C
- \[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{\cos x}{\sqrt{3}} \right) + C\]
- \[- \frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
- \[\frac{1}{2 \sqrt{3}} \tan^{- 1} \left( \frac{2 \cos x}{\sqrt{3}} \right) + C\]
- \[- e^x \tan\frac{x}{2} + C\]
- \[- e^x \cot\frac{x}{2} + C\]
- \[- \frac{1}{2} e^x \tan\frac{x}{2} + C\]
- \[- \frac{1}{2} e^x \cot\frac{x}{2} + C\]
- \[\frac{- e^{- x}}{e^x + e^{- x}} + C\]
- \[- \frac{1}{e^x + e^{- x}} + C\]
- \[\frac{- 1}{\left( e^x + 1 \right)^2} + C\]
- \[\frac{1}{e^x - e^{- x}} + C\]
2 loge cos (xex) + C
sec (xex) + C
tan (xex) + C
tan (x + ex) + C
- \[\frac{1}{3} \tan^2 x + C\]
- \[\frac{1}{2} \tan^2 x + C\]
- \[\frac{1}{3} \tan^3 x + C\]
none of these
The primitive of the function \[f\left( x \right) = \left( 1 - \frac{1}{x^2} \right) a^{x + \frac{1}{x}} , a > 0\text{ is}\]
- \[\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
- \[\log_e a \cdot a^{x + \frac{1}{x}}\]
- \[\frac{a^{x + \frac{1}{x}}}{x} \log_e a\]
- \[x\frac{a^{x + \frac{1}{x}}}{\log_e a}\]
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)
\[\int\sqrt{\frac{x}{1 - x}} dx\] is equal to
- \[\sin^{- 1} \sqrt{x} + C\]
- \[\sin^{- 1} \left\{ \sqrt{x} - \sqrt{x \left( 1 - x \right)} \right\} + C\]
- \[\sin^{- 1} \left\{ \sqrt{x \left( 1 - x \right)} \right\} + C\]
- \[\sin^{- 1} \sqrt{x} - \sqrt{x \left( 1 - x \right)} + C\]
ex f (x) + C
ex + f (x)
2ex f (x)
ex − f (x)
The value of \[\int\frac{\sin x + \cos x}{\sqrt{1 - \sin 2x}} dx\] is equal to
- \[\sqrt{\sin 2x} + C\]
- \[\sqrt{\cos 2x} + C\]
± (sin x − cos x) + C
± log (sin x − cos x) + C
tan x − x + C
x + tan x + C
x − tan x + C
− x − cot x + C
\[2\left( \sin x + x\cos\theta \right) + C\]
- \[2\left( \sin x - x\cos\theta \right) + C\]
\[2\left( \sin x + 2x\cos\theta \right) + C\]
- \[2\left( \sin x - 2x\cos\theta \right) + C\]
\[ \frac{1}{5x} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C\]
\[ \frac{1}{5} \left( 4 + \frac{1}{x^2} \right)^{- 5} + C\]
\[ \frac{1}{10x} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C\]
- \[ \frac{1}{10} \left( \frac{1}{x^2} + 4 \right)^{- 5} + C\]
\[\int\frac{x^3}{\sqrt{1 + x^2}}dx = a \left( 1 + x^2 \right)^\frac{3}{2} + b\sqrt{1 + x^2} + C\], then
\[ a = \frac{1}{3}, b = 1\]
\[a = - \frac{1}{3}, b = 1\]
\[ a = - \frac{1}{3}, b = - 1\]
- \[ a = \frac{1}{3}, b = - 1\]
\[ x + \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 - x \right| + C\]
\[ x + \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 - x \right| + C\]
\[ x - \frac{x^2}{2} - \frac{x^3}{3} - \log\left| 1 + x \right| + C\]
- \[ x - \frac{x^2}{2} + \frac{x^3}{3} - \log\left| 1 + x \right| + C\]
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]
\[\int\frac{1}{\sqrt{x} + \sqrt{x + 1}} \text{ dx }\]
\[\int\frac{1 - x^4}{1 - x} \text{ dx }\]
\[\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }\]
\[\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }\]
\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]
\[\int\sin x \sin 2x \text{ sin 3x dx }\]
\[\int\text{ cos x cos 2x cos 3x dx}\]
\[\int\sqrt{\frac{1 - x}{x}} \text{ dx}\]
\[\int\frac{1}{4 \sin^2 x + 4 \sin x \cos x + 5 \cos^2 x} \text{ dx }\]
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
\[\int\frac{x^3}{\sqrt{x^8 + 4}} \text{ dx }\]
\[\int\frac{1}{2 - 3 \cos 2x} \text{ dx }\]
\[\int\frac{1}{\sin^4 x + \cos^4 x} \text{ dx}\]
\[\int \sec^4 x\ dx\]
\[\int {cosec}^4 2x\ dx\]
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
\[\int\frac{1}{2 + \cos x} \text{ dx }\]
\[ \int\left( 1 + x^2 \right) \ \cos 2x \ dx\]
Solutions for 19: Indefinite Integrals
RD Sharma solutions for Mathematics [English] Class 12 chapter 19 - Indefinite Integrals
Shaalaa.com has the CBSE, Karnataka Board PUC Mathematics Mathematics [English] Class 12 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 Mathematics [English] Class 12 CBSE, Karnataka Board PUC 19 (Indefinite 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 Mathematics [English] Class 12 chapter 19 Indefinite Integrals are Definite Integrals, Integrals of Some Particular Functions, Some Properties of Indefinite Integral, Integration Using Trigonometric Identities, Introduction of Integrals, Evaluation of Definite Integrals by Substitution, Properties of Definite Integrals, Methods of Integration: Integration by Substitution, Integration as an Inverse Process of Differentiation, Geometrical Interpretation of Indefinite Integrals, Methods of Integration: Integration Using Partial Fractions, Methods of Integration: Integration by Parts, Fundamental Theorem of Calculus, Indefinite Integral Problems, Comparison Between Differentiation and Integration, Indefinite Integral by Inspection, Definite Integral as the Limit of a Sum, Evaluation of Simple Integrals of the Following Types and Problems.
Using RD Sharma Mathematics [English] Class 12 solutions Indefinite 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 Mathematics [English] Class 12 students prefer RD Sharma Textbook Solutions to score more in exams.
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