Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 32A / b / Solution

Let $\Phi (t)=2t^4-t^2$. Its stationary points on $[0,1]$ are the endpoint $0$ and the interior point $1/2$, withThe endpoint stationary-phase contribution to ${\int }_0^1{e}^{ix\Phi (t)}dt$ isand the interior contribution isThe nonstationary endpoint contributes only $O(x^{-1})$. Taking imaginary parts gives