Mallows' integral and some generalizations (Q1121410)

Almost ten years ago Mallows proposed the problem of directly evaluating the integral \[ \int^{\pi}_{0}[\sin u(\sin \alpha u)^{-\alpha}(\sin \beta u)^{-\beta}]^ t du, \] when \(\alpha +\beta =1\). Its value is \(\Gamma (t+1)/\Gamma (\alpha t+1)\Gamma (\beta t+1)\), so it is clearly a type of beta integral. Mallows' method of deriving this was to solve a probability problem two ways. A second evaluation using Lagrange inversity was found by Evans, Ismail and Stanton. The present paper derives this and extensions using a change of variables in Cauchy's beta integral. Related integrals are evaluated using this method and some contour bending.