\rm\int_(0)^{ (\pi)/(2) } \frac{1}{ \sqrt{1 - {sin}^(2) ( \frac{1}2) {sin}^(2) \varphi } } d \varphi \\​

Question

`\rm\int_(0)^{ (\pi)/(2) } \frac{1}{ \sqrt{1 - {sin}^(2) ( \frac{1}2) {sin}^(2) \varphi } } d \varphi \\`​

Answer 1

This is an another elliptical integral, but of the first kind:

`\displaystyle F(k) = \int_0^(\pi/2) (dx)/(√(1-k^2\sin^2(x)))`

`\implies \displaystyle \int_0^(\pi/2) (d\varphi)/(√(1-\sin^2\left(\frac12\right)\sin^2(\varphi))) = \boxed{F\left(\sin\left(\frac12\right)\right)}`