Calculate simple integral involving DiracDelta?

Questions
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1

Integrate[E^(I (x - a) t), {t, -Infinity, Infinity}]

simply returns the result \int_{-\infty }^{\infty } e^{\,i\, t\,(x\,-a\,)} \, dt

Why doesn’t it yield something like 2 \pi DiracDelta[x-a]?

2

In[3]:= FourierTransform[Sqrt[2 Pi], t, (x - a)]

Out[3]= 2 \[Pi] DiracDelta[-a + x]

works better

3

Thanks very much!

Is there a way to make Mathematica remember this rule easily, so that when it’s asked to evaluate an integral like

\int {\text d}^3x\ e^{\ \pm \ {\vec A}.{\vec x} }

so the result is

(2 \pi)^3 DiracDelta[\vec A] ?

4

I think you could if you define a new Integral function of your own and check if your integral is a fourier transform before passing it on to the regular internal integral