I am trying to solve the following integral: $$ \int_0^1 x^c (1-x)^d dx $$ for some $c, d \in \mathbb{R}$ I know I have to use the gamma function, I have tried using the substitutions $u = ln\frac{1}{x}$ as well as $u = ln\frac{1}{1-x}$ to try to make this appear as a gamma function but neither have gotten me closer to an answer.
2026-03-26 16:26:53.1774542413
Integrating $x^c (1-x)^d$ from 0 to 1 using the gamma function
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Hint:
$$\int_0^1 t^{x-1}(1-t)^{y-1}dt=B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$