Question

\[\int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}\]

Answer 1

\[\text{ Let I } = \int\frac{\sin 2x}{a^2 + b^2 \sin^2 x}dx\]
\[\text{ Putting  a }+ b^2 \sin^2 x = t\]
\[ \Rightarrow b^2 \left( 2 \sin x \cos x \right) dx = dt\]
\[ \Rightarrow b^2 \times \text{ sin  2x  dx} = dt\]
\[ \therefore I = \frac{1}{b^2}\int\frac{dt}{t}\]
\[ = \frac{1}{b^2}\text{ ln }\left| t \right| + C \]
\[ = \frac{1}{b^2}\text{ ln }\left| a^2 + b^2 \sin^2 x \right| + C ................\left( \because t = a + b^2 \sin^2 x \right)\]