Let $$\vec F = \dfrac{A}{r^2}\hat r$$
I want to calculate $$\lim_{(x_0,y_0,z_0) \to \infty}\int^{(a,b,c)}_{(x_0,y_0,z_0)} \vec F \cdot d\vec l$$
In spherical coordinates,
$$\lim_{r_2 \to \infty}\int^{r_1}_{r_2} \vec F \cdot d\vec l = \dfrac{A}{r_1}$$
I want to do the same integral in cartesian coordinates. I tried following path,
First, from $(x_0,y_0,z_0)$ to $(a, y_0, z_0)$ for $x_0,y_0, z_0 \to \infty$
Second, from $(a,y_0,z_0)$ to $(a, b, z_0)$ for $x_0,y_0, z_0 \to \infty$
Third, from $(a,b,z_0)$ to $(a, b, c)$ for $x_0,y_0, z_0 \to \infty$
I want to calculate $$\int^{(a,b,c)}_{(x_0,y_0,z_0)} \vec F \cdot d\vec l =\int^{(a,y_0,z_0)}_{(x_0,y_0,z_0)} \vec F \cdot d\vec x + \int^{(a,b,z_0)}_{(a,y_0,z_0)} \vec F \cdot d\vec y + \int^{(a,b,c)}_{(a,b,z_0)} \vec F \cdot d\vec z $$, for $x_0, y_0, z_0 \to \infty$
First integral :
$$\int^{(a,y_0,z_0)}_{(x_0,y_0,z_0)} \vec F \cdot d\vec x = \int^{(a,y_0,z_0)}_{(x_0,y_0,z_0)} \dfrac{1}{x^2 + y_0^2 + z_0^2} d x = \ \int^{(a,y_0,z_0)}_{(x_0,y_0,z_0)} \dfrac{1}{x^2 + \infty + \infty} d x $$
huh ?
How do I calculate the integral in cartesian coordinates ?
Just so that people don't start calling me out not being mathematically accurate; I want to calculate the integral in cartesian coordinates because in my Physics class, instructor calculated electrostatic work done using spherical coordinates.
For a physicist the first and second part of your chosen path take, in the limit, place infinitely far away from the origin. Hence there is no force working and both integrals have a zero contribution. Only the last integral, where you move from infinitely far away in the $z$-direction toward the point $(a,b,c)$ you integrate over a non-zero force and you will find the answer you are looking for.
As a mathematician you would work out all of the integrals for explicitly finite values of $(x_0,y_0,z_0)$ and only after having the result take the proper limits.
In either case the contribution of the integral in the last line will go to zero.
There is however an slight issue and that is concerned with the force calculation in the last line. The force has a magnitude and a direction, which is not properly included. You need to use $\vec{F} = |F| \hat{F}$ for a radial force, which gives on this particular path: $$ \vec{F} = \frac{A}{(x^2+y_0^2+z_0^2)} \frac{1}{\sqrt{x^2+y_0^2+z_0^2}} \left( \begin{array}{l} x \\ y_0 \\ z_0 \end{array}\right) $$ and hence the integral should read: $$ \int_{x_0}^a \vec{F} \cdot d \vec{x} = \int_{x_0}^a \frac{x}{(x^2+y_0^2+z_0^2)^{\frac{3}{2}}} ~dx $$ and something similar for the other two paths.
Note that for the result you want to obtain should not depend on the path you take. You therefore strictly speaking only need one of the three coordinates $(x_0,y_0,z_0)$ to go to infinity and the other two can remain finite.