Question

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

Answer 1

$\int \frac{\sin x}{(1+\sin x)(2+\sin x)}dx$
$\int \frac{2\sin x\cos x}{(1+\sin x)(2+\sin x)}dx$
Let $\{\sin x=t;\cos xdx=dt\}$
$=\int \frac{2t}{(1+t)(2+t)}dt$
$=\int (\frac{-2}{1+t}+\frac{9}{2+t})dt$
$=-2\log (1+t)+4\log (2+t)+C$
$=-2\log |\sin x+1|+4\log (\sin x+2)+C$