I want to evaluate the integral
$ \int_0^{2\pi} \sqrt{1+a\cdot \cos(x)} ~dx$ where $|a| \leq 1$.
I have already tried to split the integral using the periodicity of $\cos$ into
$ \int_0^{\pi} \left( \sqrt{1+a\cdot \sin(x)} + \sqrt{1-a\cdot \sin(x)} \right) ~dx$,
but that does not seem to make things easier.
Do you have any suggestion on how to proceed?
Best Answer
You are entering the world of elliptic integrals.
Start using $\cos(x)=1-2 \sin ^2\left(\frac{x}{2}\right)$ to make $$I=\int \sqrt{1+a\, cos(x)}\,dx=\int\sqrt{(1+a)-2 a \sin ^2\left(\frac{x}{2}\right)}\,dx$$ $$I=\sqrt{1+a}\int\sqrt{1-\frac{2 a }{1+a}\sin ^2\left(\frac{x}{2}\right)}\,dx$$ Now, let $x=2y$ $$I=2\sqrt{1+a}\int\sqrt{1-\frac{2 a }{1+a}\sin ^2\left(y\right)}\,dy$$ Have a look here.