$\int \limits_0^{\frac{\pi}{2}} \sin ^3 x \sin 2x dx$
$\qquad= \int \limits_0^{\frac{\pi}{2}} 2 \sin ^4 x \cos x dx$
$\sin x =t$
$\cos x dx=dt$
$\int \limits _0^1 2 t^4 dt$
$\qquad= \large\frac{2t^5}{5} \bigg]_0^1 $
$\qquad=\large\frac {2}{5}$
Hence b is the correct answer.