Solution
= Solution
The Gaussian decay makes the <integral> <absolutely convergent>, so <odd> symmetry gives $\boxed{\int_{-\infty}^{\infty}f(x)dx=0}$. On the positive half-line substitute $u=3x^2/2$, so $x^3dx=2u\,du/9$. Thus
$$
\boxed{\int_0^\infty f(x)dx=\frac2{27}\int_0^\infty ue^{-u}du=\frac2{27}.}
$$
The last <integral> is $\Gamma(2)=1$, an instance of the <odd Gaussian moments on a half-line>.