Solution
= Solution
For $n\geq2$, write $x^ne^{-x^2}=x^{n-1}(xe^{-x^2})$ and apply <integration by parts>, using the <antiderivative> $-e^{-x^2}/2$ for the second factor:
$$
I(n)=\left[-\frac12x^{n-1}e^{-x^2}\right]_0^\infty+\frac{n-1}{2}\int_0^\infty x^{n-2}e^{-x^2}\,dx.
$$
The boundary term is zero: at zero, $n-1>0$, and at infinity <Gaussian decay dominates every power>. Consequently the <Gaussian moment recurrence> is
$$
\boxed{I(n)=\frac{n-1}{2}I(n-2),\qquad n\geq2.}
$$