Question

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

Answer 1

$$\begin{gathered} \mathrm{Let},I={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{x}{{\sin }^{2}x+{\cos }^{2}x}dx \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{x}{1}dx \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}xdx \\ ={\left[\frac{x^2}{2}\right]}_0^{\frac{\mathrm{\pi }}{2}} \\ \therefore I=\frac{{\mathrm{\pi }}^2}{8} \end{gathered}$$