Question

Evaluate: $\int \frac{\sin 2x}{({\mathrm{asin}}^{2}x+b{\cos }^{2}x)}dx$

Answer 1

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

Put $y=a^2{\sin }^{2}x+b^2{\cos }^{2}x$

$\Rightarrow dy=(2a^2\sin x\cos x-2b^2\cos x\sin x)dx$

$=(a^2\sin 2x-b^2\sin 2x)dx$

$\Rightarrow \sin 2xdx=\frac{dy}{a^2-b^2}$

$\Rightarrow I=\frac{1}{a^2-b^2}\log y+c$

$=\frac{\log (a^2{\sin }^{2}x+b^2{\sin }^{2}x)}{a^2-b^2}+c$