Answer
$\frac{1}{2}\sin^4 x$
Work Step by Step
$\int \sin^2 x\sin 2x\ dx$
$=\int \sin^2 x*2\sin x\cos x\ dx$
$=2\int\sin^3 x\cos x\ dx$
We use u substitution. Let $u=\sin x$. Then, $du=\cos x\ dx$.
$=2\int u^3\ du$
$=2*\frac{u^4}{4}$
$=\boxed{\frac{1}{2}\sin^4 x}$