Question

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

• A. $\frac{64}{231}$
• B. $\frac{24}{231}$
• C. $\frac{54}{231}$
• D. None of these

Answer 1

The correct option is A $\frac{64}{231}$

Let

$\begin{array}{c}I={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^5\varphi d\varphi \\ ={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\cos }^4\varphi \cos \varphi d\varphi \\ ={\int }_0^{\frac{\pi }{2}}\sqrt{\sin \varphi }{\left(1-{\sin }^2\varphi \right)}^2\cos \varphi d\varphi \\ putt=\sin \varphi \\ \frac{dt}{d\varphi }=\cos \varphi \\ dt=\cos \varphi d\varphi \\ When\varphi =0,t=0when\varphi =\frac{\pi }{2},t=1\\ \therefore I={\int }_0^1\sqrt{t}{\left(1-t^2\right)}^{2}dt\\ ={\int }_0^1\sqrt{t}\left(1-2t^2+t^4\right)dt\\ ={\int }_0^1{t}^{\frac{1}{2}}+t^{\frac{9}{2}}-2t^{\frac{5}{2}}dt\\ ={\left[\frac{2}{3}{t}^{\frac{3}{2}}+\frac{2}{11}{t}^{\frac{11}{2}}-\frac{4}{7}{t}^{\frac{7}{2}}\right]}_0^1\\ =\frac{2}{3}+\frac{2}{11}-\frac{4}{7}\\ =\frac{154+42-132}{3\times 11\times 7}=\frac{64}{231}\end{array}$