Equal-argument beta integral
= Equal-argument beta integral
{title2=$B(n,n)=2^{1-2n}B(n,1/2),\quad n>0$}
In $B(n,n)=\int_0^1[x(1-x)]^{n-1}dx$, put $v=2x-1$. Then $x(1-x)=(1-v^2)/4$, and <evenness> gives
$$
B(n,n)=2^{2-2n}\int_0^1(1-v^2)^{n-1}dv.
$$
The further substitution $u=v^2$ turns this into $2^{1-2n}B(1/2,n)$. The symmetry of the <Euler beta function> yields the title identity. This proof works for every positive real $n$, without requiring $n$ to be an integer or using the <Gamma function>.