alternate integral representation of beta function

By making the change of variable $x^p=y$, we see that

$${\int }_0^1{x}^{p-1}{(1-x)}^{q-1}𝑑x=\frac{1}{p}{\int }_0^1{(1-y^{\frac{1}{p}})}^{q-1}𝑑y.$$

Hence, we have

$${\int }_0^1{(1-y^{\frac{1}{p}})}^{q-1}𝑑y=p\frac{\mathrm{\Gamma }(p)\mathrm{\Gamma }(q)}{\mathrm{\Gamma }(p+q)}.$$

Title	alternate integral representation of beta function
Canonical name	AlternateIntegralRepresentationOfBetaFunction
Date of creation	2013-03-22 17:10:08
Last modified on	2013-03-22 17:10:08
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	4
Author	rspuzio (6075)
Entry type	Result
Classification	msc 33B15