Question

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

Answer 1

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

$={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^4\varphi \cos \varphi d\varphi$

$={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{(1-{\sin }^2\varphi )}^2\cos \varphi d\varphi$

Let $t=\sin \varphi \Rightarrow dt=\cos \varphi d\varphi$

When $\varphi =0\Rightarrow t=0$ and When $\varphi =\frac{\pi }{2}\Rightarrow t=1$

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

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

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

$={\int }_0^1(t^{\frac{1}{2}}+t^{\frac{9}{2}}-2t^{\frac{5}{2}})dt$

$={[\frac{t^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{t^{\frac{9}{2}+1}}{\frac{9}{2}+1}-2\frac{t^{\frac{5}{2}+1}}{\frac{5}{2}+1}]}_0^1$

$={[\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+\frac{t^{\frac{11}{2}}}{\frac{11}{2}}-2\frac{t^{\frac{7}{2}}}{\frac{7}{2}}]}_0^1$

$={[\frac{2}{3}{t}^{\frac{3}{2}}+\frac{2}{11}{t}^{\frac{11}{2}}-\frac{4}{7}{t}^{\frac{7}{2}}]}_0^1$

$=[\frac{2}{3}(1-0)+\frac{2}{11}(1-0)-\frac{4}{7}(1-0)]$

$=\frac{2}{3}+\frac{2}{11}-\frac{4}{7}=\frac{154+42-132}{231}=\frac{64}{231}$