Integral arising from Brownian motion question I want to show that $\int_0^{\infty}exp({-a^2 / {2t} - \lambda t})\frac{a}{\sqrt{2\pi t^3}} dt = exp(-a \sqrt{2 \lambda})$. Please can you give me a clue on how to do this. I have tried integration by parts, and know that I will have to use properties of the normal density at some point, but am not having much luck.

Hint you can relate this integral to the gamma function by making suitable change of variables.
