Transforming an integral equation using taylor series 
Can I transform the equation using a taylor series, as the integration seems to impossible to be done by itself? 
 A: I think it is better to go one step further to write the integrand as:
$$
\int \int \sqrt{\left(\frac{s}{2}\right)^2+\left( R+ s \cos \frac{t}{2} \right)^2} ds dt
$$
Integral on $s$ can be taken (using Mathematica/Wolfram Alpha for example). Then see if you can simplify it more.
Another way that I can think of is changing coordinates to polar coordinates which is of course messy!
A: If $R$ is sufficiently large then we can rewrite the integrand to do a Talyor expansion. We start by pulling a factor of $R^2$ out of the square root. 
$$ \sqrt{\frac{s^2}{4}+R^2+2Rs\cos(t/2)+s^2\cos^2(t/2)}$$
$$= R \sqrt{1+\color{red}{\frac{2s\cos(t/2)}{R}+\frac{s^2/4+s^2\cos^2(t/2)}{R^2}}}$$
$$= R \sqrt{1+\color{red}{\epsilon}}=R(1+\frac12\epsilon-\frac32\epsilon^2+\cdots)$$
As indicated abouve, we can expand $\sqrt{1+\epsilon}$ in a Mclauren series. Each of the terms of this series can be integrated, though it would be a pain in the neck to do so. The series will only converge if $|\epsilon| < 1$; this requirement leads us to a lower bound for $R$. We will use the fact that $0<s<1$, $|\cos(t)|<1$, and $R>1$.
$$|\epsilon| = \frac{2s|\cos(t/2)|}{R}+\frac{s^2/4+s^2\cos^2(t/2)}{R^2} < \frac{2}{R} + \frac{3/4}{R^2} < \frac{7/4}{R}$$
So we have at least that the series will converge if $R>7/4$, though we could find tighter lower bound than this. 
