Why Doesn't Integrate Do Fourier Transforms?

Questions
1

In[2]:= FourierTransform[1, t, [Omega]]

Out[2]= Sqrt[2 \[Pi]] DiracDelta[\[Omega]]

In[5]:= 1/Sqrt[2 \[Pi]]
  Integrate[Exp[I \[Omega] t], {t, -\[Infinity], \[Infinity]}]

During evaluation of In[5]:= Integrate::idiv: Integral of E^(I t \[Omega]) does not converge on {-\[Infinity],\[Infinity]}.
2

Integrate will apparently only enter the land of generalized functions (DiracDelta and relatives) if the integrand contains a generalized function. This heuristic can be annoying.