Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 19C / d / Solution

Let $x_1,\dots ,x_N$ be the distinct real roots of ${\mathrm{He}}_N$. The Gauss-Hermite quadrature rule iswhere $w_j$ is the weighted integral of the $j$th Lagrange interpolation polynomial. Equivalently,The orthogonality of ${\mathrm{He}}_N$ makes the rule exact for every polynomial of degree at most