# Evaluate $\int\frac{(-x_{0}+R\cos(\theta))\,d\theta}{\left((-x_{0}+R\cos(\theta))^2+(-R\sin(\theta))^2\right)^{3/2}}$

As the title suggests, how may I start evaluating the following integral:

\begin{align}I=\displaystyle\int\dfrac{\left(-x_{0}+R\cos\left(\theta\right)\right)\,d\theta}{\left((-x_{0}+R\cos(\theta))^2+(-R\sin(\theta))^2\right)^{3/2}},\;\text{for }\color{red}{x_{0}>R}.\end{align}

It appeared while trying to find the resultant electric field in a plane containing a circular ring of charge. My Physics professor suggested that an elliptic integral would pop up, is this that elliptic integral? I don't know which substitution to make to make it look more obvious.

• Are $x_0$ and $R$ postive real numbers, so $x_0,R\in\mathbb{R}^+$? – Jan Feb 14 '16 at 20:17
• @JanEerland Yes, that's the assumption I built the integrand on. – jm324354 Feb 14 '16 at 20:21
• It looks like there will be elliptic functions in the primitive, yes. If you have limits on the integral, you might have a chance (I've not tried any cases) to get an elementary answer... – mickep Feb 15 '16 at 7:30