Fourier Analysis – FourierTransform Frequency

One of the methods is to adjust the FourierParameters

  FourierTransform(1, x, w, FourierParameters -> {0, -2*Pi})

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  FourierTransform(Exp(I a x), x, w, FourierParameters -> {0, -2*Pi})

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Compare

funs = {1, DiracDelta(x), Exp(I a x), Cos(a x), Sin(a x)};
result= {#, FourierTransform(#, x, w, FourierParameters -> {0, -2*Pi})}& /@ funs;
Prepend(result, {"f(x)","Fourier transform unitary, ordinary frequency"});
Grid(%, Frame -> All)

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With the second column of Wiki:

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