I'm computing a quantity using its Fourier transform. However, I don't think what I'm getting makes sense. This is what I have
$$ \Delta E = \int_{-\infty}^{+\infty} \frac{dE}{dt} dt = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} i \omega Ẽ(\omega) e^{i \omega t} \frac{d \omega}{2 \pi} dt = \int_{-\infty}^{+\infty}i \omega Ẽ(\omega) \delta(\omega) = 0. $$
What am I doing wrong?