2. $\int \sqrt{x}\sin ^2 x^{3/2}\ dx=\left\{ u=x^{3/2},\ du=\frac{3}{2}\sqrt{x}\ dx \implies dx=\frac{2du}{3\sqrt{x}} \right\}=\int \frac{2}{3}\sin^2 u \ du$
Use the identity $\sin^2 u = \frac{1-\cos 2u}{2}$