Let $\boldsymbol{l}:\mathbb{R}\to\mathbb{R}^3$ be the piecewise smooth parametrization of an infinitely long curve $\gamma$. Let us define$$\boldsymbol{B}(\boldsymbol{x})=\int_\gamma\frac{d\boldsymbol{l}\times(\boldsymbol{x}-\boldsymbol{l})}{\|\boldsymbol{x}-\boldsymbol{l}\|^3}=\int_{-\infty}^{+\infty}\frac{\boldsymbol{l}'(t)\times(\boldsymbol{x}-\boldsymbol{l}(t))}{\|\boldsymbol{x}-\boldsymbol{l}(t)\|^3}dt.$$A physical interpretation of the integral is that $\boldsymbol{B}$ represents the magnetic field generated by an infinitely long wire $\gamma$ carrying a current $I$ such that $\mu_0 I=4\pi$ (where $\mu_0$ is vacuum permeabilty).
Can we be sure that the integral converges and, if we can, how can it be proved?
I think that the best way to approach the problem is verifying whether the integral converges as a Lebesgue integral, which is equivalent to verifying whether the integral of the absolute value of the integrand converges, and I notice that every component of the integral is the difference of two terms having the form $l_i'(t)(x_j-l_i(t))\|\boldsymbol{x}-\boldsymbol{l}(t)\|^{-3}$. I see that $|l_i'(t)(x_j-l_i(t))|\|\boldsymbol{x}-\boldsymbol{l}(t)\|^{-3}$ $\le |l_i'(t)||x_j-l_i(t)|^{-2}$, but the absolute value does not allow me to use the rule $l_i'(t)dt=dl_i$...