Beta function identity

Question

How to prove the equality $${ {4}^{s}}\ B\left(s+{1\over2},{1\over2}\right)={2 } \int_{0}^{2}x^{2s}(4-x^2)^{-1/2}dx, $$ where B(x,y) represents the beta function.

Answer 1

The substitution $x=2\sqrt{y}$ and $dx=\frac1{\sqrt{y}}dy$ turn the RHS into $$\int_0^2x^{2s}(4-x^2)^{-1/2}dx=2^{2s-1}\int_0^1y^{s-1/2}(1-y)^{-1/2}dv.$$