Improper line integral (${\lim }_{\epsilon \downarrow 0}{\int }_{a+\epsilon }^b\mathbf{F}(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt$)

A line integral can be defined as an improper integral of its parametrized integrand when that integrand is undefined or unbounded at isolated parameter values. At an interior singular parameter, split the integral and require both one-sided limits to exist separately; their cancellation in a symmetric Cauchy principal value is a different convention. For a scalar arc length integral, use $f(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|$ in place of the vector-field dot product.

If the restricted integrand has a removable singularity, integrating its continuous extension gives the same value. This does not require the original vector field to extend continuously to a full neighbourhood of the curve. Likewise, a differential identity may hold only on the regular segments: a zero integrand there has improper integral zero, although its printed classical expression may not be defined at the omitted points.