Question

Solve:
${\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^5\varphi d\varphi$

Answer 1

$I={\int }_0^{\pi /2}\sqrt{\sin \varphi }{\cos }^5\varphi d\varphi$.

Let $\sin \varphi =t$. Then, $\cos \varphi d\varphi =dt$

Also, $\varphi =0\Rightarrow t=\sin 0=0$ and $\varphi =\frac{\pi }{2}\Rightarrow t=\sin \frac{\pi }{2}=1$

$\therefore I={\int }_0^1\sqrt{t}(1-t^2{)}^{2}dt={\int }_0^1\sqrt{t}(1-2t^2+t^4)dt$

$\Rightarrow I={[\frac{2}{3}{t}^{3/2}-\frac{4}{7}{t}^{7/2}+\frac{2}{11}{t}^{11/2}]}_0^1=\frac{2}{3}-\frac{4}{7}+\frac{2}{11}=\frac{64}{231}$