Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y)

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Theorem

The following holds for $b>-1$, $\mathrm{Re}(x)>0$, and $\mathrm{Re}(y)>0$: $$\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}(1+bt)^{-x-y} \mathrm{d}t = (1+b)^{-x} B(x,y),$$ where $B$ denotes the beta function.

Proof

References

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