I'm trying to IFT this fraction after I couldn't find it in a Fourier couples table.

\(\displaystyle F(j\omega)=\frac{250\pi \delta(\omega)}{30+125\omega}\).

When I integrated the latter, I got: \(\displaystyle \frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30+125\omega}e^{-j\omega t}d\omega\) \(\displaystyle \ =\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30}d\omega\) \(\displaystyle =\frac{25}{6} u(t)\)??

Is there ever a \(\displaystyle u(t)\) in IFT ??

Thanks for any help!