Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y)

Theorem

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

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (17)$