Gauss-Hermite quadrature exploits the fact that quadrature approximations are open integration
formulas (that is, the values of the endpoints are not required) to evaluate of integrals in the
range \((-\infty, \infty )\).
The weighting function in this case is \(\exp(-x^2)\), which results in the evaluation points
being roots of Hermite polynomials.
Therefore, the method can be used for finding the integral
\[
\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx.
\]