Question

$\underset{0}{\overset{\mathrm{\pi }/2}{\int }}\frac{{\sin }^{3/2}x}{{\sin }^{3/2}x+{\cos }^{3/2}x}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathit{}\mathit{}I={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}d\mathrm{x}...\left(\mathrm{i}\right) \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{\mathrm{n}}\left({\displaystyle \frac{\mathrm{\pi }}{2}}-\mathrm{x}\right)}{{\sin }^{\mathrm{n}}\left(\frac{\mathrm{\pi }}{2}-\mathrm{x}\right)+{\cos }^{\mathrm{n}}\left(\frac{\mathrm{\pi }}{2}-\mathrm{x}\right)}d\mathrm{x}\mathrm{Using}{\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{\mathrm{a}}\mathrm{f}\left(\mathrm{a}-\mathrm{x}\right)\mathrm{dx} \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^{\mathrm{n}}\mathrm{x}}{{\cos }^{\mathrm{n}}\mathrm{x}+{\sin }^{\mathrm{n}}\mathrm{x}}\mathrm{dx} \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\cos }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}\mathrm{dx}...\left(\mathrm{ii}\right) \\ \mathrm{Adding}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\mathrm{we}\mathrm{get} \\ 2I={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}+\frac{{\cos }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}\mathrm{dx} \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}\mathrm{dx} \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\mathrm{dx} \\ ={\left[\mathrm{x}\right]}_0^{\frac{\mathrm{\pi }}{2}}=\frac{\mathrm{\pi }}{2} \\ \mathrm{Hence}I=\frac{\mathrm{\pi }}{4} \\ \mathrm{i}.\mathrm{e}.,{\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{\mathrm{n}}\mathrm{x}}{{\sin }^{\mathrm{n}}\mathrm{x}+{\cos }^{\mathrm{n}}\mathrm{x}}d\mathrm{x}=\frac{\mathrm{\pi }}{4} \\ \therefore {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{{\sin }^{3/2}\mathrm{x}}{{\sin }^{3/2}\mathrm{x}+{\cos }^{3/2}\mathrm{x}}d\mathrm{x}=\frac{\mathrm{\pi }}{4} \end{gathered}$$