First note that, by Fubini theorem, the Fourier transform equals: for every ,

(you may have a slightly different definition of the Fourier transform; there are several conventions)

This shows that it suffices to find the Fourier transform of the function from to .

There are various proofs for this, and I think you should have been given hints to find it.

A possibility is to show that the Fourier transform satisfies a first order differential equation. Here are the steps:

Differentiate (with respect to ) and integrate by parts (by integrating and dividing ). You get .

Hence there is such that .

To find the value of , remark that . And (this can be proved by polar change of variable if you don't know that). As a consequence, . (There may be mistakes here)

Notice that if , the function is equal to its own Fourier transform.