Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 6A / b / Solution

A two-node Gaussian rule is exact through degree three, the maximal degree $2n-1$. Its nodes are the zeros of the monic quadratic orthogonal to $1,x$ for weight $x$ on $[0,1]$. Writing it as $p=x^2+ux+v$ givesso $u=-6/5$, $v=3/10$. Therefore