This question was asked in my mid term exam which has been concluded and I was unable to solve it and couldn't do it again when I tried to do at home. Hence , I am posting it here for help.
Prove that $\frac{ \Gamma(p) \Gamma(p)} { \Gamma(2p)} = 2 \int_{0}^{1/2} x^{p-1} (1-x)^{p-1}dx$.
I am not able to think how this problem should be attempted as I am not able to get any intution regarding it. So, I request you to give some hints so that I am able to start it.
Obviously the integrand is symmetric about $x=\frac12$, so the factor of $2$ may be merged into the integral bounds giving $$\int_0^1x^{p-1}(1-x)^{p-1}\,dx$$ which by definition of the beta function is $$\mathrm B(p,p)=\frac{\Gamma(p)^2}{\Gamma(2p)}$$