I would like to ask if anyone would help me to explain how to reach the following relation. \begin{equation} \int_0^1 \left( 1-x^{\frac{1}{p}} \right)^q dx= \frac{p!\,q!}{(p+q)!} \end{equation} If we substitute one half for p and q \begin{equation} \int_0^1 \sqrt{ 1-x^2} dx = \frac{\pi}{4} \end{equation} then \begin{equation} \frac{p!\,q!}{(p+q)!} \rightarrow (1/2)!^2 \rightarrow (1/2)! = \frac{\sqrt{\pi}}{2} \end{equation} I just proved that one half of the factorial is $\frac{\sqrt{\pi}}{2}$.
Thank you in advance.
Do the substitution $x=t^p$ to get \begin{eqnarray*} p \int_0^1 t^{p-1}(1-t)^q dt. \end{eqnarray*} This is the beta function ... https://en.wikipedia.org/wiki/Beta_function