Given $\displaystyle \displaystyle a,b\in\mathbb{R} $, compute $\displaystyle \displaystyle \int_0^\infty e^{-ax^2}\cos(bx)dx $.
Is...
$\displaystyle \displaystyle \int_{0}^{\infty} e^{-a\ x^{2}}\ \cos b x\ dx = \frac{1}{2} \ \mathcal{F} \{e^{-a\ x^{2}}\} = \sqrt{\frac{\pi}{4\ a}}\ e^{-\frac{b^{2}}{4a}}$
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$