Evaluating $\int_0^\infty e^{-u^2}\cos(xu)du$ How do I evaluate $$\int_0^\infty e^{-u^2}\cos(xu)du$$
Any hints?

One may differentiate under the integral sign: let $J(x) = \int_0^{\infty} e^{-u^2}\cos{xu} \, du$. We have
$$ J'(x) = -\int_0^{\infty} ue^{-u^2}\sin{xu} \, du $$
Integrating by parts,
$$ J'(x) = \left[\frac{1}{2} e^{-u^2}\sin{xu} \right]_0^{\infty} - \frac{x}{2}\int_0^{\infty} e^{-u^2}\cos{xu} \, du, $$
so $J'(x) = -\frac{x}{2}J(x)$. We also know $J(0) = \sqrt{\pi}/2$, and solving the differential equation gives
$$ J(x) = \frac{\sqrt{\pi}}{2} e^{-x^2/4}. $$
