$\sqrt {\sin 2x} \cos 2x$

: Let $I = \int {\sqrt {\sin 2x} \cos 2xdx}$
Put $\sin 2x = t$ $\Rightarrow$ $2\cos 2xdx = dt$

$\therefore$ $I = \cfrac{1}{2}\int {{t^{1/2}}dt} = \cfrac{1}{2} \cdot \cfrac{{{t^{\cfrac{1}{2} + 1}}}}{{^{\cfrac{1}{2} + 1}}} + C$

$= \cfrac{1}{2} \times \cfrac{2}{3}{t^{3/2}} + C = \cfrac{1}{3}{t^{3/2}} + C = \cfrac{1}{3}{\left( {\sin 2x} \right)^{3/2}} + C$