We know that in 1d, $\frac{1}{|x|^{3.5}}$ is not locally integrable around the origin. I learnt from this post that we could apply the following definition to define a tempered distribution:
$$\langle\frac{1}{|x|^{3.5}},\phi\rangle:=\int _{-\infty}^{\infty}\frac{\phi(x)-\sum_{m=0}^{2}\frac{\phi^{(m)}(0)}{m!}x^m}{|x|^{3.5}}dx,\quad \phi\in\mathbb{S}.$$
My question is, here are we actually considering the Cauthy principal value of the above integral?
If so, how do we actually show $\langle\frac{1}{|x|^{3.5}},\phi\rangle$ defined above is a well-defined tempered distribution?
Thank you very much in advance for any of your helps:-)