I've been asked to evaluate

[tex]F(\omega) = \frac{1}{\sqrt{2\pi}}\int dte^{-\alpha t}cos\omega t[/tex]

and I've not the foggiest idea what to do. I thought I could just go about doing in the integral by parts (limits are 0 and infinity by the way), but on further research I don't think I can do that can I?