Question

Evaluate the integral ${\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^5\varphi d\varphi$ using substitution.

Answer 1

Let $I={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }co{s}^5\varphi d\varphi ={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^4\varphi d\varphi$
Also, let $\sin \varphi =t\Rightarrow \cos \varphi d\varphi =dt$
When $\varphi =0,t=0$ and when $\varphi =\frac{\pi }{2},t=1$
$\therefore I={\int }_0^1\sqrt{t}(1-t^2{)}^{2}dt$
$={\int }_0^1{t}^{\frac{1}{2}}(1+t^4-2t^2)dt$
$={\int }_0^1[t^{\frac{1}{2}}+t^{\frac{9}{2}}-2t^{\frac{5}{2}}]dt$
$={[\frac{t^{\frac{3}{2}}}{\frac{3}{2}}+\frac{t^{\frac{11}{2}}}{\frac{11}{2}}-\frac{2t^{\frac{7}{2}}}{\frac{7}{2}}]}_0^1$
$=\frac{2}{3}+\frac{2}{11}-\frac{4}{7}$
$=\frac{154+42-132}{231}=\frac{64}{231}$