By changing the order of integration, prove that ∫_0^1▒∫_0^(√(1-x^2 ))▒〖{dy/((1+e^y)√(1-x^2-y^2 ))}dx=π/2 log⁡(2e/(1+e)) 〗

Question

By changing the order of integration, prove that ∫_0^1▒∫_0^(√(1-x^2 ))▒〖{dy/((1+e^y)√(1-x^2-y^2 ))}dx=π/2  log⁡(2e/(1+e)) 〗

Steve · Answer

Is the integrand really         1 ------------------------------- (1+e^y)√(1-(x^2+y^2)) ?? I don't think that's doable with elementary functions