$\int\limits_{-\infty}^\infty {{e^{i\omega t}}d\omega} = 2\pi \delta \left( t \right)$ is generally accepted.
But
Integrate[E^(I w t), {w, -\[Infinity], \[Infinity]},
Assumptions -> t \[Element] Reals]
gives
Integrate::idiv: Integral of E^(I t w) does not converge on {-[Infinity],[Infinity]}.
and
Integrate[E^(I w t), {w, -\[Infinity], \[Infinity]},
PrincipalValue -> True, Assumptions -> t \[Element] Reals]
gives 0
So is it possible to get the correct delta function result?
DiracDelta
say: "Integrate never gives DiracDelta as an integral of smooth functions:... FourierTransform can give DiracDelta", there are appropriate examples, as well. $\endgroup$