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 f(x)&=&x. I want to perform a Fourier transform on f(x) 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
\tilde{f}(\omega)&=&\frac{1}{\sqrt{2\pi}}\int f(x)e^{i\omega x}dx,
and the inverse Fourier transform is
f(x)&=&\frac{1}{\sqrt{2\pi}}\int\tilde{f}(\omega)e  ^{-i\omega x}d\omega.
What I did was the following: The Fourier transform of f(x)&=&x is given by
\tilde{f}(\omega)&=&\frac{1}{\sqrt{2\pi}}\int xe^{i\omega x}dx,
then performing an inverse Fourier transform on this new function yields
f(x)=\frac{1}{\sqrt{2\pi}}\int\left[\frac{1}{\sqrt{2\pi}}\int xe^{i\omega x}e^{-i\omega x}dxd\omega\right].
Now, the e terms become 1 and I'm assuming dx cancels with d\omega, so we are left with
f(x)=\frac{1}{2\pi}\int\int x,
which is clearly not the original function I started with. So, does anyone know what rookie error I made to go so horribly wrong?