Advertisements
Advertisements
Question
\[\int\frac{1}{\sin^2 x + \sin 2x} \text{ dx }\]
Sum
Advertisements
Solution
\[\text{ Let I } = \int \frac{1}{\sin^2 x + \sin \left( 2x \right)}dx\]
\[ = \int \frac{1}{\sin^2 x + 2 \sin x \cos x}dx\]
\[\text{Dividing numerator and denominator by} \cos^2 x\]
\[ \Rightarrow I = \int \frac{\sec^2 x}{\tan^2 x + 2 \tan x}dx\]
\[\text{ Let tan x } = t\]
\[ \Rightarrow \sec^2 x \text{ dx } = dt\]
\[ \therefore I = \int \frac{dt}{t^2 + 2t}\]
\[ = \int \frac{dt}{t^2 + 2t + 1 - 1}\]
\[ = \int \frac{dt}{\left( t + 1 \right)^2 - \left( - 1 \right)^2}\]
\[ = \frac{1}{2}\text{ ln } \left| \frac{t + 1 - 1}{t + 1 + 1} \right| + C\]
\[ = \frac{1}{2}\text{ ln } \left| \frac{t}{t + 2} \right| + C\]
\[ = \frac{1}{2}\text{ ln } \left| \frac{\tan x}{\tan x + 2} \right| + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\sqrt{x}\left( 3 - 5x \right) dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int\frac{1}{1 - \cos x} dx\]
\[\int \cos^{- 1} \left( \sin x \right) dx\]
\[\int\frac{1}{\left( 7x - 5 \right)^3} + \frac{1}{\sqrt{5x - 4}} dx\]
\[\int\frac{1}{\sqrt{x + 1} + \sqrt{x}} dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
\[\int \cos^2 \text{nx dx}\]
\[\int\frac{\sec x \tan x}{3 \sec x + 5} dx\]
\[\ ∫ x \text{ e}^{x^2} dx\]
\[\int\frac{x^4 + 1}{x^2 + 1} dx\]
\[\int\frac{\sec^2 x}{\sqrt{4 + \tan^2 x}} dx\]
\[\int\frac{x}{x^2 + 3x + 2} dx\]
\[\int\frac{x^3}{x^4 + x^2 + 1}dx\]
\[\int\frac{\left( 1 - x^2 \right)}{x \left( 1 - 2x \right)} \text
{dx\]
\[\int\frac{1}{\cos x \left( \sin x + 2 \cos x \right)} dx\]
\[\int x \cos x\ dx\]
\[\int x \sin x \cos x\ dx\]
\[\int\frac{\sin^{- 1} x}{x^2} \text{ dx }\]
\[\int\frac{x \sin^{- 1} x}{\sqrt{1 - x^2}} dx\]
\[\int \sin^3 \sqrt{x}\ dx\]
\[\int\sqrt{x^2 - 2x} \text{ dx}\]
\[\int\sqrt{2x - x^2} \text{ dx}\]
\[\int\frac{1}{\left( x + 1 \right) \sqrt{x^2 + x + 1}} \text{ dx }\]
\[\int\frac{1}{1 + \tan x} dx =\]
\[\int\frac{\sin^2 x}{\cos^4 x} dx =\]
\[\int\frac{\sin x}{1 + \sin x} \text{ dx }\]
\[\int \text{cosec}^2 x \text{ cos}^2 \text{ 2x dx} \]
\[\int\frac{\left( \sin^{- 1} x \right)^3}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\frac{e^x - 1}{e^x + 1} \text{ dx}\]
\[\int x \sin^5 x^2 \cos x^2 dx\]
\[\int \sin^5 x\ dx\]
\[\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} \text{ dx}\]
\[\int \tan^5 x\ \sec^3 x\ dx\]
\[\int x^2 \tan^{- 1} x\ dx\]
\[\int\frac{e^{m \tan^{- 1} x}}{\left( 1 + x^2 \right)^{3/2}} \text{ dx}\]
\[\int\frac{x^2 + 1}{x^2 - 5x + 6} \text{ dx }\]
