I'd like to numerically integrate $\frac{1}{\sqrt{f(x)}}$ on an interval between two consecutive zeros of the function $f(x)$, which makes the integrand singular at two endpoints. Standard practice seems to be to split into intervals so that each has one singularity at an endpoint. However, I don't know the locations of these endpoints explicitly; at best, I can solve for them numerically to arbitrary precision. This presents a dilemma because one needs to take into account the size of the singularity (i.e. the size of $f'(x)$ at the endpoints) to know how precisely to specify the endpoint. What's the best way to handle such a situation? Can standard software carry out the whole procedure to arbitrary precision?
Numerical quadrature when locations of singularities are approximate
numerics;quadrature