I get that the integral $\int^{+\infty}_{-\infty} t^{-2} e^{i\omega t} dt $ is not convergent, since there is a singularity at $t =0$.
However, I know (Wolfram) that Fourier transform of $f(x) = t^{-2}$ is:
$ g(\omega)=-\sqrt{\frac{\pi}{2}}\omega$
Can somebody explain how is this possible?