I've stumble into this integral on a paper (page 5, right column)
$\displaystyle\int_{\tfrac {\cos(k)}{\sqrt{1+\cos^2(k)}} \leq~y}~ \dfrac{\mathrm{d}k}{4\pi}$
with solution
$\displaystyle{1-\frac{1}{\pi}\arccos \left(\frac{k}{\sqrt{1-k^2}}\right)}$.
What type of integral is this? How would you solve it? How do the authors of this paper solve this?