Fourier Integral and the Dirac Delta

This is going to end up being a simple question with a simple answer, but I just can't tweak the answer out of my Math Methods text.

I have the following statement:

$\displaystyle \dot{\Delta}(\vec{x},0) = i \int \frac{d^3k}{2(2 \pi )^3} \left [ e^{i \vec{k} \cdot \vec{x}} + e^{-i \vec{k} \cdot \vec{x}} \right ] $ ($\displaystyle \vec{k}$ and $\displaystyle \vec{x}$ are real 3-vectors and $\displaystyle \vec{k} \cdot \vec{x}$ is the usual dot product.)

I'm supposed to get that

$\displaystyle \dot{\Delta}(\vec{x},0) = -i \delta ^3 (\vec{x})$

But looking at the integral I'm thinking it ought to be:

$\displaystyle \dot{\Delta}(\vec{x},0) = \frac{i}{2} \left ( \delta ^3 (\vec{x}) + \delta ^3 (-\vec{x}) \right )$

which would give me 0, which obviously isn't true. So (sigh) what am I doing wrong?

Thanks!

-Dan