Hilbert Transform in mathematica

Question

I would like to do a Hilbert transformation in Mathematica on a function. However, it does not seem to give a right result.

The Hilbert transform is given by

So I did :

 Integrate[Sin[t]/(k - t), {t, -Infinity, Infinity}];

but it does not work

The only reference I found online is this :

https://www.wolframalpha.com/input?i=Hilbert+transform+sinx

Does anyone have an idea on how to do this transform as a Mathematica code?

Thanks