Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi \,d \phi$.

Let $I = \int_{0}^{\frac{\pi}{2}} \sqrt{\sin \phi} \cos ^{5} \phi \,d \phi$.
Substitute $\sin \phi = t$,then $\cos \phi \,d \phi = dt$.
When $\phi = 0$,$t = 0$. When $\phi = \frac{\pi}{2}$,$t = 1$.
Since $\cos^4 \phi = (1 - \sin^2 \phi)^2 = (1 - t^2)^2$,the integral becomes:
$I = \int_{0}^{1} \sqrt{t} (1 - t^2)^2 \,dt$
$I = \int_{0}^{1} t^{\frac{1}{2}} (1 + t^4 - 2t^2) \,dt$
$I = \int_{0}^{1} (t^{\frac{1}{2}} + t^{\frac{9}{2}} - 2t^{\frac{5}{2}}) \,dt$
Integrating term by term:
$I = \left[ \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + \frac{t^{\frac{11}{2}}}{\frac{11}{2}} - 2 \frac{t^{\frac{7}{2}}}{\frac{7}{2}} \right]_{0}^{1}$
$I = \left[ \frac{2}{3} t^{\frac{3}{2}} + \frac{2}{11} t^{\frac{11}{2}} - \frac{4}{7} t^{\frac{7}{2}} \right]_{0}^{1}$
$I = \frac{2}{3} + \frac{2}{11} - \frac{4}{7}$
Finding a common denominator $(231)$:
$I = \frac{2(77) + 2(21) - 4(33)}{231} = \frac{154 + 42 - 132}{231} = \frac{64}{231}$.