Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 14A / b / iii / Solution

For the weight $w=x$, substitution $x=e^s$ givesThe product-to-sum formula makes this zero when $m\ne n$ and $1/2$ when $m=n$. ThereforeIn particular the eigenfunctions are orthogonal, and multiplying each by $\sqrt{2}$ gives an orthonormal basis for the weighted space.