${\int }_0^{2\pi } \frac{x{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx,n>0,$ is equal to

Let $I={\int }_0^{2\pi } \frac{x{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx$                                    …(i)

Using property IV, we have 

$\begin{array}{l}{I}_n={\int }_0^{2\pi } \frac{(2\pi -x){\sin }^{2n}(2\pi -x)}{{\sin }^{2n}(2\pi -x)+{\cos }^{2n}(2\pi -x)}dx\\ \Rightarrow I_n={\int }_0^{2\pi } \frac{(2\pi -x){\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx\end{array}$

Adding (i) and (ii), we get

$\begin{array}{l}2I_n=2\pi {\int }_0^{2\pi } \frac{{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx\\ \Rightarrow I_n=\pi {\int }_0^{2\pi } \frac{{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}\mid x}dx\\ \Rightarrow I_n=2\pi {\int }_0^{\pi } \frac{{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx \text{ [Using property VII] }\\ \Rightarrow I_n=4\pi {\int }_0^{\pi /2} \frac{{\sin }^{2n}x}{{\sin }^{2n}x+{\cos }^{2n}x}dx \text{ [Using property VII] }\\ \Rightarrow I_n=4\pi \times \frac{\pi }{4}={\pi }^2\\ \end{array}$