This is related to a separate post: Question about computing a Fourier transform of an integral transform related to fractional Brownian motion, but because the question is so basic, I thought I would give it some different tags.
How do you show the Fourier transform with repsect to $x$ of $ 2 \int_{0}^{\infty} \frac{e^{\frac{-x^2}{2y}}}{\sqrt{2 \pi y}} e^{-2y}\mathrm dy$ is $\frac{4}{k^2+4}$ ?