∫ 1 Cos X ( Sin X + 2 Cos X ) D X - Mathematics

Question

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

Sum

Solution

\[\text{ Let I }= \int \frac{1}{\cos x\left( \sin x + 2 \cos x \right)}dx\]
\[\text{Dividing numerator and denominator by} \cos^2 x\]
\[ \Rightarrow I = \int \frac{\sec^2 x}{\frac{\cos x}{\cos x} \times \left( \frac{\sin x + 2 \cos x}{\cos x} \right)}dx\]
\[ = \int \frac{\sec^2 x}{\left( \tan x + 2 \right)}dx\]
\[\text{ Let tan x } + 2 = t\]
\[ \Rightarrow \sec^2 x \text{ dx } = dt\]
\[ \therefore I = \int \frac{dt}{t}\]
\[ = \text{ ln } \left| t \right| + C\]
\[ = \text{ ln } \left| \tan x + 2 \right| + C\]

shaalaa.com

Indefinite Integral Problems

Is there an error in this question or solution?

Q 9 Q 8 Q 10

Chapter 19: Indefinite Integrals - Exercise 19.22 [Page 114]

APPEARS IN

RD Sharma Mathematics [English] Class 12

Chapter 19 Indefinite Integrals
Exercise 19.22 | Q 9 | Page 114

Video TutorialsVIEW ALL [1]

view
Video Tutorials For All Subjects
Indefinite Integral Problems

video tutorial00:39:37

RELATED QUESTIONS

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

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

\[\int \left( 2x - 3 \right)^5 + \sqrt{3x + 2} \text{dx} \]

\[\int\frac{1 - \cot x}{1 + \cot x} dx\]

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

\[\int\frac{\sin\sqrt{x}}{\sqrt{x}} dx\]

\[\int\frac{e^{3x}}{4 e^{6x} - 9} dx\]

\[\int\frac{x}{3 x^4 - 18 x^2 + 11} dx\]

\[\int\frac{\sin 8x}{\sqrt{9 + \sin^4 4x}} dx\]

` ∫ \sqrt{"cosec x"- 1} dx `

\[\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }\]

`int 1/(sin x - sqrt3 cos x) dx`

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

\[\int\frac{\log \left( \log x \right)}{x} dx\]

\[\int \left( \log x \right)^2 \cdot x\ dx\]

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

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

\[\int \sin^{- 1} \left( 3x - 4 x^3 \right) \text{ dx }\]

\[\int \sin^3 \sqrt{x}\ dx\]

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

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

\[\int x^2 \sqrt{a^6 - x^6} \text{ dx}\]

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

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

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

\[\int\frac{x^2 + 9}{x^4 + 81} \text{ dx }\]

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

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

\[\int\frac{\cos2x - \cos2\theta}{\cos x - \cos\theta}dx\] is equal to

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

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

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

\[\int\sqrt{\sin x} \cos^3 x\ \text{ dx }\]

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

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

\[\int\frac{1}{5 - 4 \sin x} \text{ dx }\]

\[\int x^3 \left( \log x \right)^2\text{ dx }\]

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

\[\int\frac{\sin 4x - 2}{1 - \cos 4x} e^{2x} \text{ dx}\]