TITLE: Applications of integrals of rational functions of sine and cosine
QUESTION [6 upvotes]: I earlier asked this question about conformal equivalence of flat tori with embedded tori.
In the ensuing thread the integral $\displaystyle\int\frac{dx}{R+\cos x}$ occurred.  If I'm not mistaken, it was for just that integral that Euler first introduced the tangent half-angle substitution sometimes (probably incorrectly?) called the Weierstrass substitution.
So there is an application to geometry, of an integral of a rational function of sine and/or cosine.
$\Large\mathbb Q$uestion: What other applications of such integrals exist?

REPLY [3 votes]: To get the total cross section, you also have to integrate over the azimuthal angle. Since there is no azimuthal dependence, that will just result in a factor of $$ 2\pi $$ to the total cross section, for a final result of $$ \sigma = \frac{a^2}{2}\cdot2\pi = \pi a^2$$ This is also the cross-sectional area of the scattering potential.