Evaluate: ∫ (sin x dx)/(cos x(1 - sin x))

$\frac{\sin x}{\cos x(1-\sin x)} = \frac{\sin x(1+\sin x)}{\cos x\cdot\cos^2 x} = \tan x\sec^2 x + \tan^2 x\sec x$. $\int \tan x\sec^2 x\,dx = \frac{\tan^2 x}{2}$. $\int \tan^2 x\sec x\,dx = \int \sec^3 x\,dx - \int \sec x\,dx = \frac{1}{2}\sec x\tan x + \frac{1}{2}\ln|\sec x + \tan x| - \ln|\sec x + \tan x|$. Hence the integral $= \frac{1}{2}\tan^2 x + \frac{1}{2}\sec x\tan x - \frac{1}{2}\ln|\sec x + \tan x| + C$.