Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 1 / c / ii / Solution

Write the integral's multiplier in the Chebyshev polynomial basis:At ${\gamma }^2=5$, part (i) givesBy orthogonality, the $y_1$ term and all cross terms vanish. Using $I_0=\pi$ and $I_2=\pi /2$, the integral isThe remainder can be integrated against this integrable weight in the regular expansion, so it retains the stated order.