This paper presents the analytic integration of the free-space Green's function over rectangular regions arising in electromagnetic theory. Specifically, closed-form formulas are presented for the integration of 1/R and 1/R² functions over a linear segment, over a rectangular surface, and over a rectangular volume. Presented formulas can be used for the calculation of the electrostatic potential, the electric field strength, the magnetic vector potential, and the magnetic field strength due to source (charge or current) uniformly distributed over rectangular regions.