My Beta Functions - Part 2

Let the given integral be $I$

Take the substitution $x^p = t ..... (1)$

Differentiate $(1)$

$dx px^{p-1} = dt \Rightarrow dx p \frac{x^p}{x} = dt \Rightarrow dx p \frac{t}{t^{\frac{1}{p}}} = dt \Rightarrow dx p t^{\frac{p-1}{p}} = dt \Rightarrow dx = \dfrac{dt}{p t^{\frac{p-1}{p}}}$

$I = \displaystyle \frac{1}{p} \int _{0}^{1} \dfrac{t^{\frac{m-1}{p}}}{t^{\frac{p-1}{p}}} (1-t)^{n-1} dt$

$\displaystyle \Rightarrow \frac{1}{p} \int_{0}^{1} t^{\frac{m-p}{p}} (1-t)^{n-1} dt$

$\displaystyle \Rightarrow \frac{1}{p} \int _{0}^{1} t^{\frac{m}{p}-1} (1-t)^{n-1} dt$

Which is $\frac{1}{p} \beta(\frac{m}{p} ,n)$