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So, as you suggested, I added a numerical Definition for f_0, now I'm getting an error:

x,w,f_0,t = var("x,w,f_0,t")

f_0 = 1
w = 2*pi*f_0

x(t) = sin(w*t)

integrate(x*exp(-I*w*t),t, -oo, oo)


This is the actual result:

Traceback (click to the left of this block for traceback)
...
ValueError: Integral is divergent.