Question

Solve:$\int \frac{\cos xdx}{(1+\sin x)(2+\sin x)}$

Answer 1

$\int \frac{\cos xdx}{(1+\sin x)(2+\sin x)}=I$ (let)
Let $\sin x=t$, then $\cos xdx=dt$
$\therefore I=\int \frac{1}{(1+t)(2+t)}dt$
$=\int (\frac{1}{1+t}-\frac{1}{2+t})dt$
$=\ln |1+t|-\ln |2+t|+C$
$=\ln |\frac{1+t}{2+t}|+C$
$=\ln |\frac{1+\sin x}{2+\sin x}|+C$