Fourier transforms and their inverse

I'm having a little bit of trouble that I know should be trivial but for some reason I cannot get it. As a simple example consider . I want to perform a Fourier transform on and then perform an inverse Fourier transform on this transformed function to get back to the original function. Now, the Fourier transform is given by

,

and the inverse Fourier transform is

.

What I did was the following: The Fourier transform of is given by

,

then performing an inverse Fourier transform on this new function yields

.

Now, the terms become 1 and I'm assuming cancels with , so we are left with

,

which is clearly not the original function I started with. So, does anyone know what rookie error I made to go so horribly wrong?