The Pareto principle (a.k.a. 80/20 rule) says that for many outcomes, roughly 80% of consequences come from 20% of the causes. Qualitatively speaking, this also applies to numerical integration: most of the integral can be handled by just a few quadrature rules.

People who have learned numerical analysis all know about the Gauss quadrature rule: for any integer $n>0$, there exist $n$ nodes $x_1,x_2,\dots,x_n\in[-1,1]$ and $n$ corresponding weights $w_1,w_2,\dots,w_n$, such that the approximation

is highly accurate for any function $f$ smoothly defined on $[-1,1]$. The error of this approximation typically decays exponentially as $n$ increases. Together with scaling and shifting of variables, the Gauss quadrature efficiently handles all the regular integrals (i.e. integrals involving smooth integrands) on any interval $[a,b]\subset\mathbb{R}$.

What if the integrand is singular? How would you approximate an integral when the integrand is not smooth or even blowing up somewhere on the interval? It turns out that most of the singular integrals in practice can be handled by a few strategies (80/20 rule again!). Here they are: