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 / i / Solution

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.