Assume is harmonic, then it has a harmonic conjugate such that is analytic and satisfies the partials:

so yes, you can partially integrate them like:

and after the integration with respect to y with x held constant, we obtain something like:

where the constant of integration is now a function of x right? Now we use the second CR equations to solve for by writing:

Now, solve for and integrate.

Do it manually a few times first to get the hang of it then explain how that's the same as:

where all the integrations are without the arbitrary function, and it looks intimidating at first sight but becomes a piece of cake after you solve a few of them. For example , then:

Now take the derivative of that with respect to x:

and continue working the partials and integrals until you get the expression for v(x,y). Then the most general harmonic conjugate is v(x,y)+k.