1. ## cosine and exponential

$\frac{1}{\sqrt{2\pi t}} \int_{-\infty}^{\infty} cos(ax) e^{-x^2/2t} dx$

2. Originally Posted by graticcio

$\frac{1}{\sqrt{2\pi t}} \int_{-\infty}^{\infty} cos(ax) e^{-x^2/2t} dx$

$\int^\infty_{-\infty} e^{\frac{{-x^2}}{2}} dx = \sqrt{2\pi}$

I think if you carry out integration by parts using this you should find that the integral is equal to zero. Does this help?