Question

Evaluate \[\int\limits_{0}^{1}[{{1-x^2 \over 1+x^2}]^{1/2}}dx\] In terms of the Gamma function

Answer 1

The limits suggest trying to get at this through the beta function,
\[ B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} \ dt = \frac{ \Gamma(x) \Gamma(y)}{\Gamma(x+y)} \]
But I can't see right now how to transform this into a combination of B(x,y). I'd be interested in seeing the solution when you have it.