$$\vec{F}_{net} = \frac{Gm_pm_o}{2\pi}\int_0^{2\pi}\frac{(R\cos\theta - r_o)\hat{i}+R\sin\theta\hat{j}}{(R^2+r_o^2-2Rr_o\cos\theta)^\frac{3}{2}}d\theta$$
Is this integral possible? (Here $R, r_o$ are constants)
Thanks for the help in advance!
P.S. I'll soon make a post in physics stack exchange on how I got to this in the first place.
Edit: I was able to solve the second term of the integral i.e. the $\hat{j}$ term, but still am not able to resolve the $\hat{i}$ term.
The $i$-integral is to be expressed in term of the elliptical function $K(t)$ \begin{align} \int_0^{2\pi}\frac{R\cos\theta - r}{(R^2+r^2-2Rr\cos\theta)^{3/2}}d\theta =& \frac d{dr} \int_0^{2\pi}\frac{1}{(R^2+r^2-2Rr\cos\theta)^{1/2}}d\theta\\ =& \frac d{dr} \bigg(\frac{2K\left(\frac{4Rr}{(R+r)^2}\right)}{R+r} + \frac{2K\left(-\frac{4Rr}{(R-r)^2}\right)}{R-r} \bigg) \end{align}