= Solution
The <principal-value reciprocal distribution> is
$$
\left\langle\operatorname{pv}\frac1x,\varphi\right\rangle
=\lim_{\varepsilon\downarrow0}\int_{|x|>\varepsilon}\frac{\varphi(x)}x\,dx.
$$
The limit exists because the constant part of the numerator cancels symmetrically at zero. Multiplying by $x$ gives $x\operatorname{pv}(1/x)=1$.
If $v$ is any other solution, $w=v-\operatorname{pv}(1/x)$ obeys $xw=0$. To identify this <kernel of multiplication by a coordinate>, choose a <cutoff function> $\eta$ equal to one near zero. Every test function has the form $\varphi=\varphi(0)\eta+x\psi$ with $\psi\in\mathcal D$. Thus $\langle w,\varphi\rangle=\varphi(0)\langle w,\eta\rangle$, proving $w=c\delta_0$. Therefore
$$
\boxed{v=\operatorname{pv}\frac1x+c\delta_0,\qquad c\in\mathbb C.}
$$
For real distributions the constants are real.
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