Numerical Integration for integrable singularity

With an open quadrature method such as Gauss-Legendre you may not need to evaluate the integrand at the point of singularity. However, proceeding this way will most likely result in a severe loss of accuracy,

If you are lucky, you may be able to remove the singularity with a change of variables. For example if $f \in C([0,1])$ and $0 < \alpha < 1/2$ then the improper integral

$$\int_0^1 \frac{f(x)}{x^\alpha} \, dx,$$ can be transformed under the change of variables $t = x^\alpha$ to

$$\frac{1}{\alpha}\int_0^1 f(t^{1/\alpha})t^{(1 - 2\alpha)/\alpha}\, dt,$$

which is a proper integral and can be handled efficiently by the techniques you mentioned.

More generally, an understanding of the asymptotic behavior of the integrand near the singularity is important.

The standard numerical methods are designed for smooth (or at least sufficiently differentiable) functions, and don't perform very well in the presence of a singularity. One thing that may work well is to subtract out the singularity. That is, if you want $\int_a^b f(x)\; dx$, try to write $f(x) = g(x) + h(x)$ where $g(x)$ has the singularity but is integrable in closed form, while $h(x)$ has no singularity. See e.g. these notes of mine.

There are a few different methods to deal with this problem. Gaussian quadrature can work in some cases, you then absorb the singularity in a suitably chosen weight function and then you proceed in the usual way with constructing the orthogonal polynomials where the inner product is defined using that weight function.

Another method is to do a change of variables to eliminate the singularity as mentioned in RRL's answer, or you can split the integrand in a singular part and a non-singular part as mentioned in Robert Israel's answer.

A general purpose method that doesn't require much effort to implement is the Tanh-sinh quadrature method, here you transform an integral from minus 1 to 1 to one over the entire real line, singularities at the endpoints are then not going to cause problems.