How to prove:

$\int_{0}^{\frac{\pi }{2}} {\frac{\sin \theta}{\sqrt{Z^2+(R+h \tan \theta)^2}} K[k(\theta)]}=\frac{\pi }{2\sqrt{R^2 + (h+Z)^2}} $

where \[ k(\theta)=\sqrt\frac{4Rh \tan \theta}{Z^2+(R+h \tan \theta)^2}\]

and $ K[k(\theta)] $ is the complete elliptic integral of the first kind, defined by

\[ K[k(\theta)]= \int_0^{\frac{\pi }{2}}\frac{\,d\phi}{\sqrt{1-k^2(\theta)\sin^2 \phi}}\]

and h, R and Z $ \gt 0 $