Question

$\int \frac{\sin 2x}{a^2+b^2{\sin }^{2}x}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{\sin 2x}{a^2+b^2{\sin }^{2}x}dx \\ \mathrm{Putting}{\sin }^{2}x=t \\ \Rightarrow 2\sin x.\cos x=\frac{dt}{dx} \\ \Rightarrow \sin 2x=\frac{dt}{dx} \\ \Rightarrow \sin 2xdx=dt \\ \therefore I=\int \frac{1}{a^2+b^{2}t}dt \\ =\frac{1}{b^2}\ln \left|a^2+b^{2}t\right|+C \\ =\frac{1}{b^2}\ln \left|a^2+b^2{\sin }^{2}x\right|+C\left[\because t={\sin }^{2}x\right] \end{gathered}$$