Question

Integrate the rational function $\frac{\cos x}{(1-\sin x)(2-\sin x)}$

Answer 1

Let $\sin x=t\Rightarrow \cos xdx=dt$
$\therefore \int \frac{\cos x}{(1-\sin x)(2-\sin x)}dx=\int \frac{dt}{(1-t)(2-t)}$
Let $\frac{1}{(1-t)(2-t)}=\frac{A}{(1-t)}+\frac{B}{(2-t)}$
$\Rightarrow 1=A(2-t)+B(1-t)$ ............. (1)
Substituting $t=2$ and then $t=1$ in equation (1), we obtain
$A=1$ and $B=-1$
$\therefore \frac{1}{(1-t)(2-t)}=\frac{1}{(1-t)}-\frac{1}{(2-t)}$
$\Rightarrow \int \frac{\cos x}{(1-\sin x)(2-\sin x)}dx=\int \{\frac{1}{1-t}-\frac{1}{(2-t)}\}dt$
$=-\log |1-t|+\log |2-t|+C$
$=\log \left|\frac{2-t}{1-t}\right|+C$
$=\log \left|\frac{2-\sin x}{1-\sin x}\right|+C$