Compute $\int_0^1x^m(1-x^n)^pdx$

Question

Compute $\int_0^1x^m(1-x^n)^pdx$

Hint in question says, express in terms of gamma function, which I don't see how. I can put $x=\frac{1}{e^t}$ and make limit from $0$ to infinity, as in gamma function. Any other method is equally fine. Thanks.

Answer 1

Hint. As @Did has noticed, one may recall the Euler Beta integral result:

$$ B(a,b)=\int\limits_0^1 t^{a-1}(1-t)^{b-1}\mathrm{d}t=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}, \quad a>0,\,b>0. $$

Then by the change of variable $x=t^{1/n}$, $dx=\frac1n t^{1/n-1}dt$, one gets $$ \int_0^1x^m(1-x^n)^pdx=\frac1n\int_0^1t^{\large \frac{m+1}n-1}(1-t)^{p+1-1}dt. $$ Can you take it from here?