Convective heat transfer for steady state, laminar, hydrodynamically developed flow in microtubes with uniform temperature and uniform heat nux boundary conditions are solved by the integral transform technique. Temperature jump condition at the wall and viscous heating within the medium are included. The solution method is verified for the cases where viscous heating is neglected. For uniform temperature case, with a given Brinkman number, at specified axial lengths, the viscous effects are presented for the developing range, reaching the fully developed Nusselt number. The effect of viscous heating is investigated for both of the cases where the fluid is being heated or cooled. Prandtl number analysis has shown that, as we increase the Prandtl number the temperature jump effect diminishes which gives a rise to the Nusselt number.