Dipole line-integral identity (${\int }_0^c\frac{\mathbf{r}-t{\mathbf{r}}_0}{|\mathbf{r}-t{\mathbf{r}}_0{|}^3}dt=\frac{c(P_c\mathbf{r}+r{\mathbf{P}}_c)}{r{P}_c(r{P}_c+\mathbf{r}\cdot {\mathbf{P}}_c)}$)

Here $r=|\mathbf{r}|$, ${\mathbf{P}}_c=\mathbf{r}-c{\mathbf{r}}_0$ and $P_c=|{\mathbf{P}}_c|$. The identity holds where the segment of source points $t{\mathbf{r}}_0$, $0\le t\le c$, avoids the observation point and the denominator is nonzero. Differentiating the right side with respect to $c$ gives the integrand and its value at $c=0$ is zero. Dotting with a current dipole moment evaluates a continuous line of dipole electric potentials.