Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 1 / b / iii / Solution

Using the normalized eigenfunctions and the same change of variables, their weighted inner product isFor distinct positive integers $m,n$, the product-to-sum identity givesbecause each integrated sine has integer-multiple-of-$\pi$ endpoints. For $m=n$, the integral is one by the normalization already calculated. HenceThis verifies orthonormality explicitly, with ${\delta }_{mn}$ the Kronecker delta.