\[\begin{equation} {\Large\int_0^\infty} \frac{dx}{\sqrt{x} \left[x^2+\left(1+2\sqrt{2}\right)x+1\bigg] \bigg[1-x+x^2-x^3+\cdots+x^{2014}\right]} \end{equation}\]
Given that the integral above is equal to \(\pi(\sqrt a - b) \), where \(a\) and \(b\) are positive integers, find the value of \(a+b\).
by
Anastasiya Romanova
