In recent years there has been an increased emphasis on applying high order Godunov-type algorithms to the system of ideal MHD. This is motivated by their strong shock capturing and their conservation properties which make them ideally suited for use in combination with adaptive mesh refinement. Such efforts, however, have traditionally met with difficulty owing to the divergence free constraint on the magnetic field. We describe a new, unsplit MHD Godunov-type integration algorithm which uses the Constrained Transport approach to ensure the divergence free character of the magnetic field. The algorithm includes two novel features, 1) the incorporation of MHD source terms in the PPM-type reconstruction procedure and 2) an upwind CT-algorithm for combining the Godunov fluxes to calculate the electric fields needed for CT. We present test calculations comparing this algorithm against previously published results. Finally, we apply this algorithm to the study of the MRI including radiative cooling.