Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 16 / c / Solution

Write $B(n,n)={\int }_0^1[x(1-x){]}^{n-1}dx$ and set $v=2x-1$. Since $x(1-x)=(1-v^2)/4$ and $dx=dv/2$,In the last integral put $u=v^2$, giving $dv=du/(2\sqrt{u})$. This yields $B(n,n)=2^{1-2n}B(1/2,n)$, and part (a) interchanges the arguments. HenceThe equal-argument beta integral identity holds for every positive real $n$, not only positive integers.