Chapter 11.12, Problem 21P

Show that the four answers given in Section 1 for ${\displaystyle {\int }_0^{\pi /2}d}\theta /\sqrt{\cos \theta }$ are all correct. Hints: For the beta function result, use (6.4). Then get the gamma function results by using (7.1) and the various $\Gamma$ function formulas. For the elliptic integral, use the hint of Problem 17 with $\alpha =\pi /2$.