Evaluate: ∫ (sin 2x)/(1 + sin x)(2 + sin x) dx

$\sin 2x = 2\sin x\cos x$. Put $\sin x = t$, so $\cos x\,dx = dt$. $\int\frac{2t}{(1+t)(2+t)}dt$ Partial fractions: $\frac{2t}{(1+t)(2+t)} = \frac{-2}{1+t} + \frac{4}{2+t}$ $= -2\log|1+t| + 4\log|2+t| + C$ Hence the integral is $4\log(2+\sin x) - 2\log(1+\sin x) + C$.