We applied the method of finite-part integration [Galapon E.A Proc.R.Soc A 473, 20160567(2017)] to evaluate in closed-form the exact one-loop integral representations of the Heisenberg-Euler Lagrangian from QED for a constant electric field and electric-like self-dual background. We also devise a prescription based on the finite-part integration of the Cauchy principal value integral to sum and extrapolate the non-alternating divergent weak-field expansions of the Heisenberg-Euler Lagrangians to recover the non-perturbative Schwinger effect well into the strong field limit.