Fourier transform of t, 1/t and t^n

dimension10
I would like to know how one finds the Fourier transforms of

$$t$$,

$$\frac{1}{t}$$

and

$${t}^{n}$$

with the definition of the fourier transform as

$$\mathscr{F}\{f(t)\}=\mathcal{F}\{f(t)\}=\frac{1}{ \sqrt{2\pi} }\int\limits_{-\infty}^{\infty}{e}^{-i\omega t}f(t)\mbox{d}t$$

I have tried the definition of a fourier transform and I got some weird limits. Laplace transforms are so much easier!

$$\sqrt{2\pi}\delta(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty dt~e^{i\omega t}.$$
$$\sqrt{2\pi}\delta'(\omega) = \frac{i}{\sqrt{2\pi}} \int_{-\infty}^\infty dt~t e^{i\omega t}.$$
You can keep taking derivatives to get the Fourier transform of tn. For 1/t, the fourier transform will be proportional to the $\mbox{sgn}(\omega)$ function, where sgn(x) returns the sign of x.