TITLE: Evaluating $\int_0^\infty \sqrt x e^{-x^3} dx$
QUESTION [7 upvotes]: I want to evaluate:
$$\int_0^\infty \sqrt x e^{-x^3} dx$$
I guess I should use integration by parts but as I'm not good at it, I cannot go any further.

REPLY [7 votes]: Using a slightly different substitution than Peter did, $u = x^3$, we get an answer in terms of $\Gamma$:
$$
\begin{align}
\int_0^\infty \sqrt x e^{-x^3} dx
&=\int_0^\infty u^{1/6}e^{-u}\frac13u^{-2/3}\;\mathrm{d}u\\
&=\frac13\int_0^\infty u^{-1/2} e^{-u}\;\mathrm{d}u\\
&=\frac13\Gamma\left(\frac12\right)\\
&=\frac{\sqrt{\pi}}{3}
\end{align}
$$