<s>[INST] Computation of $\int_0^\infty xe^{-\frac{x^2}{2}}\sin(x \xi) \, dx$ How to compute $$\int_0^\infty xe^{-\frac{x^2}{2}}\sin(x \xi) \, dx$$
I tried to integrate by part but it was bad idea.
 [/INST] Fourier transform, contour shifting, differential equation and other nice techniques work nicely here. Since they are already explained by other users, let me show a brutal-force computation.
Using the substitution $u = x^2/2$,
\begin{align*}
\int_{0}^{\infty} x \sin(\xi x) e^{-x^2/2} \, dx
&= \int_{0}^{\infty} \sin(\xi\sqrt{2u}) e^{-u} \, du \\
&= \int_{0}^{\infty} \left( \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} \xi^{2n+1} (2u)^{n+\frac{1}{2}} \right) e^{-u} \, du \\
&= \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\xi^{2n+1} 2^{n+\frac{1}{2}} \Gamma\left(n + \frac{3}{2}\right) \\
&= \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\xi^{2n+1} 2^{n+\frac{1}{2}} \cdot \frac{\sqrt{\pi}}{2}\prod_{k=1}^{n} \left(k + \frac{1}{2}\right) \\
&= \sqrt{\frac{\pi}{2}} \sum_{n=0}^{\infty} \frac{(-1)^n}{n! 2^n}\xi^{2n+1} \\
&= \sqrt{\frac{\pi}{2}} \xi e^{-\xi^2/2}.
\end{align*}
Here, we utilized the following identities
$$ \Gamma(s) = \int_{0}^{\infty} u^{s-1}e^{-u} \, du, \qquad \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}. $$
Also, interchanging the summation and integral is justified by Fubini's theorem together with the estimation
$$\int_{0}^{\infty} \left( \sum_{n=0}^{\infty} \frac{|\xi|^{2n+1}}{(2n+1)!} (2u)^{n+\frac{1}{2}} \right) e^{-u} \, du
= \int_{0}^{\infty} \sinh(|\xi|\sqrt{2u}) e^{-u} \, du < \infty. $$
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