Find the Fourier Transform of $2x/(1+x^2)$ I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.

Another way to solve this. As you know, $$-ix \mathcal{F}[g](x) = \mathcal{F}[g'](x)$$ and $$\mathcal{F}[\mathcal{F}[g]] = cg$$
You have (with some coefficients)
$$ \mathcal{F}[ x \mathcal{F}[g](x) ] (\xi)= \mathcal{F}[i\mathcal{F}[g'] ] (\xi) = i g'(\xi)$$
Now call $f = \mathcal{F}[g]$, and you got
$$ \mathcal{F}[ x f(x) ] (\xi)= i \mathcal{F}[f ]'(\xi)$$
So if you know $\mathcal{F}[f ]$, you can conclude
(I left the exaxct value of the coefficients in the wind, there may be some nasty $2\pi$ and other values here)
