How con I make mathematica handle the complex exponential integral?

Questions
1

I would like mathematica to understand that the integral of a complex exponential is a dirac delta. Is there a way to do that? The only way I´ve thought is to use the FourierTransform.

2

Unfortunately, this is not yet built in to Mathematica. It should be.
You can calculate

In[7]:= (1/(2 Pi)) Integrate[Exp[I k (x - x0)], {k, -a, a}]

Out[7]= Sin[a (x - x0)]/([Pi] (x - x0))

The limit as a->Infinity is a well-known representation of the delta function.

3

Addendum: however

In[8]:= FourierTransform[1, k, x]

Out[8]= Sqrt[2 \[Pi]] DiracDelta[x]