Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics V / 4 / a / ii / Solution

First let $b>0$. Substitute $x=-y$ in the negative half of the integral. Since $g$ is an odd function,The two halves cancel, soFor $b=0$ both limits coincide. For $b<0$, reverse the limits and apply the result with $-b>0$. Thus the identity holds for every real $b$.