Proof of even fourier transform

My problem is:

"Prove that if f is an even function, then f^ is also an even function"

f^ = fourier transform of f."

This is what I've done, after some hints from a teaching assistant:

$\displaystyle \hat{f}(w)=\int f(x)e^{(-iwx)}dx$

$\displaystyle \hat{f}(w)=\int f(-x)e^{(-iwx)}dx$, since f(x)=f(-x), even function

Choose u=-x ->dx = -du

$\displaystyle \hat{f}(w)=-\int f(-u)e^{(-iw(-u))}du$

$\displaystyle \hat{f}(-w)=-\int f(u)e^{(-i(-w)(-u))}du$

I can't see how this answers the problem, is there anything wrong in my calculations, or do you have any hints on how to continue? I'm thankful for any help...