How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$? [duplicate]
Possible Duplicate:
How can I calculate $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?
How can we integrate $$\int_0^\frac{\pi}2\frac{\sin^nx}{\sin^nx+\cos^nx}\,\mathrm dx , \,\,\,\,\,\,\,\,\, n\in N \quad?$$ Thanks for any hint.
Hint: Make the change of variable $u=\frac{\pi}{2} -x$, noting that $\sin\left(\frac{\pi}{2}-x\right)=\cos x$. Then replace the letter $u$ by $x$, and the answer will hit you.
Remark: The hint is given in the language of formal manipulations, but the idea is purely geometric.