Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 4 / b / Solution

On bounded intervals the exponential series converges uniformly, allowing termwise integration. Since $e^{-t^2}=1-t^2+t^4/2+O(t^6)$,The requested first two nonzero terms are thereforeThe function is the error function, with $I(0)=0$ and derivative $2e^{-x^2}/\sqrt{\pi }$.