I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$
Shouldn't it be
$$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ \infty} \hat{f}(w') \, d w'?$$
I remember the Laplace transform analogue,
$$\mathcal{L} \left\{ \frac{f(t)}{t} \right\} = \int_s^{+\infty} F(s') \, ds'.$$