The value of $ \int \limits_{0}^{2\pi} \large\frac{x \sin ^{2n}x }{\sin ^{2n} x+ \cos ^{2n} x}= \pi^2$

Question

$\begin{array}{1 1} \pi^2 \\ 2 \pi^2 \\ \frac{\pi^2}{4} \\ \frac{\pi^2}{2} \end{array} $