Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 13 / b

Retain the original PDF's full-line integration limits. For $n\ge 2$, write $x^n{e}^{-x^2}dx=-x^{n-1}d(e^{-x^2})/2$. Integration by parts givesGaussian decay dominates every power, so the endpoint term vanishes, soThis is the full-line Gaussian moment recurrence; its initial values differ from the half-line version.

The integrand for $I_1$ is an odd function with absolute convergence, so $I_1=0$. For $I_2$, the recurrence gives $I_2=I_0/2$, where the supplied Gaussian integral is $I_0=\sqrt{\pi }$. HenceThe zero first moment is specific to the symmetric full-line domain.