I have problems finding the solution of this exercise:
Find the Fourier transfrom of the signal. $$ x(t) = e^{-t} \delta' $$
I have tried the following but i'm not sure if it's correct.
$$ X(w) = \int_{-\infty}^\infty e^{-t} \delta'e^{-iwt} dt$$
$$ X(w) = \int_{-\infty}^\infty \frac d {dt}\delta e^{-(1+iw)t} dt$$
Using integration by parts $$ u = e^{-(1+iw)t} \\du = \frac {-1}{1+iw}e^{-(1+iw)t} \\ v = \delta(t) \\ dv = \frac d{dt} \delta(t)$$
$$ X(w) = \delta e^{-(1+iw)t} + \int_{-\infty}^\infty \delta(t) \frac {1}{1+iw}e^{-(1+iw)t} dt $$
Evaluating the first parts becomes zero $$ X(w) = \int_{-\infty}^\infty \delta(t) \frac {1}{1+iw}e^{-(1+iw)t} dt $$
At this point i don't know how to transform this. I would like to know if my calculations are correct and how to continue.