Question

Evaluate: $\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)}dx$

Answer 1

Consider, $I=\int \frac{\sin 2x}{(1+\sin x)(2+\sin x)}$

$I=\int \frac{2\sin x\cos x}{(1+\sin x)(2+\sin x)}$

Let $\sin x=t$

Differentiate wrt $x$

$\cos xdx=dt$

$I=\int \frac{2t}{(1+t)(2+t)}$

$I=\int \frac{A}{(1+t)}+\int \frac{B}{(2+t)}$

Using method of partial fractions,we get,

$A=-2,B=4$

$I=\int \frac{-2}{(1+t)}+\int \frac{4}{(2+t)}$

$I=-2\ln |1+t|+4\ln |2+t|+C$

$I=\ln \frac{{(2+t)}^4}{{(1+t)}^2}+C$

$I=\ln \frac{{(2+\sin x)}^4}{{(1+\sin x)}^2}+C$