A closed-form expression is derived for the integration over a triangle of the normal component of electric flux from a dipole in a homogeneous space, weighted by a linear basis function. This integral arises in the finite element solution of the three-dimensional Poisson equation with a dipole source when solving for the "subtracted potential," i.e., the total potential minus the potential of the dipole in an infinite homogeneous medium. This closed-form integral evaluation allows for an efficient finite element solution for the subtracted potential that is more accurate than can be achieved using a direct finite element solution for the total potential, since the subtracted potential is a smoother function with no singularity. The formulation begins by first deriving a formula for the electric flux through a triangle due to a dipole source in a homogeneous medium (no basis function weighting). This formulation is based on an identity that is derived, which equates the electric flux from a dipole source through an arbitrary surface to a contour integral of the electric field of a point (monopole) source along the boundary of the surface. This identity results in a simple closed-form expression for the electric flux of a dipole through a triangular surface. As an extension of this result, a closed-form expression is then derived for the more complicated case where the dipolar flux density through the triangular surface is weighted by a linear basis function. Compared to numerical integration, the closed-form expression is more computationally efficient and main tains accuracy regardless of how close the dipole gets to the surface.