Question

Integrate : $\frac{\cos x}{(1-\sin x)(2-\sin x)}$

Answer 1

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

substitute $u=1-\sin x\to du=-\cos xdx$

$\int -\frac{1}{u(u+1)}dx$

$-(\int \frac{1}{u}-\frac{1}{u+1}du)$

$-(\ln u-\ln (u+1\left)\right)$

$-\ln |1-\sin x|+\ln |2-\sin x|+C$