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$\int_{0}^{1}x(1-x^2)^ndx=\frac{1}{2n+2}$ The book says a simple calculation shows this,but I don't know how to calculate. Maybe it's not simple for me...

Please anyone help.

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Let $u=1-x^2$. Then $du=-2x\,dx$, and you basically end up integrating $u^n$.

I expect you can now push things through. Please indicate if there is any difficulty.

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    Thank you very much! I calculated and got the answer!2012-11-29
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actually, it is connected to the Beta function

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    ...or elaborate a bit what you mean...2012-11-29