let, $\mathrm{z}=\sin \mathrm{x}$; $\mathrm{dz}=\cos \mathrm{x} \mathrm{dx}]$

$\begin{array}{l} \int_{0}^{\frac{\pi}{2}} \cos ^{3} x \sqrt{\sin x} d x=\int_{0}^{\frac{\pi}{2}}\left(1-\sin ^{2} x\right) \sqrt{\sin x} \cos x d x=\int_{0}^{1}\left(1-z^{2}\right) \sqrt{z} d z \\ =\left[\frac{z^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{z^{\frac{5}{2}+1}}{\frac{5}{2}+1}\right]_{0}^{1}=\left[\frac{2}{3} z^{\frac{3}{2}}-\frac{2}{7} z^{\frac{7}{2}}\right]_{0}^{1}=\frac{8}{21} \end{array}$