Mathematica Mathematica: Fourier Transform by hand of sin


i do not understand why i can find the FT of sin in mathematica using the built in function but not by integrating, even thoiugh they should be the same:

Integrate[Exp[i(ω-ω0) t,{t, -∞, ∞}, Assumptions ->ω0 el Reals && ω el Reals]

but the statment
FourierTransform[Sin[ω0 x], x, ω]
gives me the answer in dirac deltas
I Sqrt[\[Pi]/2] DiracDelta[ω - ω0] - I Sqrt[\[Pi]/2] DiracDelta[ω + ω0]

also i would be interested if you could explain to me from what condition the root 2pi appears before the FT, and how we can normalize plane waves to dirac Deltas.

Thanks a lot!!!

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