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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.

Mr Pie
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M.H
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  • I saw this question before. The first time I saw it it was a bit trickier. It said evaluate $\int_0^{\frac{\pi}{2}}\frac{1}{1+\tan^n(x)}dx$ – Amr Feb 01 '13 at 20:25
  • sir Amr you saw it but i dont saw it before i posted my question i search in this site and cant find it because these question has different form (i m not trickier) – M.H Feb 01 '13 at 20:33
  • @ Maisam Hedyelloo I didnt mean this! I saw it before in a book called "The art and craft of problem solving" (and not in this site). I was just saying a different form of the problem. – Amr Feb 01 '13 at 20:37
  • Its OK. The copy I had, I borrowed it from my university's library. – Amr Feb 01 '13 at 20:46
  • There is the same question here: http://math.stackexchange.com/questions/82489/how-can-i-calculate-int-0-pi-2-frac-sin3-t-sin3-t-cos3-tdt?rq=1 – Tomás Feb 01 '13 at 20:51
  • hi @tomas : i try to find it but i cant thanks any more – M.H Feb 01 '13 at 21:01

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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.

André Nicolas
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