Even though the integrand is not smooth, we could render it smooth by extracting from it the weight function of an orthogonal polynomial, i.e., we are rewriting $$\begin{equation} I= \int_a^b f(x)dx =\int_a^b W(x)g(x)dx \approx \sum_{i=1}^N\omega_ig(x_i), \tag{7} \end{equation}$$ where $g$ is smooth and $W$ is the weight function, which is to be associated with a given orthogonal polynomial. Note that with a given weight function we end up evaluating the integrand for the function $g(x_i)$.