The star formation rate - stellar mass relation (SFR-M*) and its evolution (i.e., the SFR main sequence) describes the growth rate of galaxies of a given stellar mass and at a given redshift. Assuming that present-day star-forming galaxies were always star-forming in the past, these growth rate observations can be integrated to calculate average star formation histories (SFHs). Using this Main Sequence Integration (MSI) approach, we trace present-day massive star-forming galaxies back to when they were 10-20% of their current stellar mass. The integration is robust throughout those epochs: the SFR data underpinning our calculations is consistent with the evolution of stellar mass density in this regime. Analytic approximations to these SFHs are provided. Integration-based results reaffirm previous suggestions that current star-forming galaxies formed virtually all of their stellar mass at z<2. It follows that massive galaxies observed at z>2 are not the typical progenitors of star-forming galaxies today. We also check MSI-based SFHs against those inferred from analysis of the fossil record -- from spectral energy distributions (SEDs) of star-forming galaxies in the Sloan Digital Sky Survey, and color magnitude diagrams (CMDs) of resolved stars in dwarf irregular galaxies. Once stellar population age uncertainties are accounted for, the main sequence is in excellent agreement with SED-based SFHs (from VESPA). Extrapolating SFR main sequence observations to dwarf galaxies, we find differences between MSI results and SFHs from CMD analysis of ACS Nearby Galaxy Survey Treasury and Local Group galaxies. Resolved dwarfs appear to grow much slower than main sequence trends imply, and also slower than slightly higher mass SED-analyzed galaxies. This difference may signal problems with SFH determinations, but it may also signal a shift in star formation trends at the lowest stellar masses.