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I was wondering if someone could possibly tell me what this integral is or possibly help me figure out an upper and lower bound for this integral (in terms of $X,Y,\alpha$):

$\int_{X}^{Y} v^{\alpha} (1-v)^{\alpha} dv$, where $-1< \alpha <0$ and $0< X < Y <1$. $\alpha, X, Y$ are all real numbers.

Thank you!

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    Google "Beta Function". It is very beautiful stuff.2012-11-11

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You can have a closed form formula in terms of the hypergeometric function

$\int_{x}^{y} v^{\alpha} (1-v)^{\alpha} dv= -{\frac {{x}^{1+\alpha}{ _2F_1(-\alpha,1+\alpha;\,2+\alpha;\,x)}-{y}^{1+ \alpha}{_2F_1(-\alpha,1+\alpha;\,2+\alpha;\,y)}}{1+\alpha}},$ where $_2F_1$ $ \,_2F_1(a,b;c;z) = \sum_{n=0}^\infty {(a)_n (b)_n \over (c)_n} \, {z^n \over n!} $

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    @JKasahara: You are welcome. Glad to assist.2012-11-11