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.

Your function is square integrable. So the Fourier transform will be square integrable, and expressed as the Cauchy principal value
$$
    \lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R}^{R}e^{-isx}\frac{2x}{1+x^{2}}dx \\
  = \lim_{R\rightarrow\infty}\frac{1}{\sqrt{2\pi}}\int_{-R}^{R}e^{-isx}\left(\frac{1}{x+i}+\frac{1}{x-i}\right)dx.
$$
Integration by parts gives evaluation terms that vanish as $R\rightarrow\infty$ so that the above becomes
$$
  = \frac{i}{s}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-isx}\left(\frac{1}{(x+i)^{2}}+\frac{1}{(x-i)^{2}}\right)dx.
$$
The Cauchy principal value limit was removed because this last integral expression is absolutely convergent.
