Fourier Transform of inverse powers of the absolute value

Question

I don't think this question has been asked previously, so here goes.
I need to evaluate the following integrals -
$$ \displaystyle{{\int\limits_{-\infty}^{\infty}}}\mathrm{d}x\dfrac{e^{-ikx}}{|x|}\text{ and }\displaystyle{\int\limits_{-\infty}^{\infty}}\mathrm{d}x\dfrac{e^{-ikx}}{|x|^3} $$ Noting that they are fourier transforms (in an appropriate convention), I plugged them into wolfram alpha, which gave me the following answers -
$$ \displaystyle{{\int\limits_{-\infty}^{\infty}}}\mathrm{d}x\dfrac{e^{-ikx}}{|x|}=-2\ln(|k|)-2\gamma $$ $$ \displaystyle{\int\limits_{-\infty}^{\infty}}\mathrm{d}x\dfrac{e^{-ikx}}{|x|^3}=\dfrac{1}{2}k^2[2\ln(|k|)+2\gamma-3] $$ Both $|x|^{-1}$ and $|x|^{-3}$ display non-integrable singularities in the conventional sense and they are not analytic (so I don't know how, if at all, these integrals would be performed in the complex x plane), yet wolfram alpha comfortably gives me an answer. So here is my question - In what sense are these integrals defined to be made evaluable to finite answers, and how am I to interpret these results? Along with this, if anyone knows how one could go about obtaining these answers, please tell me.