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How can I calculate $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?

How can we integrate $$\int_0^\frac{\pi}2\frac{\sin^nx}{\sin^nx+\cos^nx}dx , \,\,\,\,\,\,\,\,\, n\in N \quad?$$ Thanks for any hint.

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marked as duplicate by Amzoti, Brett Frankel, Micah, TMM, Thomas Feb 1 '13 at 21:50

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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 1 '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) – Maisam Hedyelloo Feb 1 '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 1 '13 at 20:37
Its OK. The copy I had, I borrowed it from my university's library. – Amr Feb 1 '13 at 20:46
There is the same question here:… – Tomás Feb 1 '13 at 20:51
up vote 18 down vote accepted

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.

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That answer is surely mean. – Pedro Tamaroff Feb 1 '13 at 20:22
Hmmmm.......... – Anon Feb 1 '13 at 20:25
Andre-nicolas your answer is very nice thanks – Maisam Hedyelloo Feb 1 '13 at 20:37

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