Integral arising from Brownian motion question

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.

Answer 1

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