Popular cosmological scenarios predict that galaxies form hierarchically from the merger of many progenitors, each with their own unique star formation history (SFH). We use a sophisticated approach to constrain the SFHs of 4517 blue (presumably star-forming) galaxies with spectroscopic redshifts in the range 0.2 < z < 1.4 from the All-Wavelength Extended Groth Strip International Survey. This consists in the Bayesian analysis of the observed galaxy spectral energy distributions with a comprehensive library of synthetic spectra assembled using realistic, hierarchical star formation, and chemical enrichment histories from cosmological simulations. We constrain the SFH of each galaxy in our sample by comparing the observed fluxes in the B, R, I, and K sub(s) bands and rest-frame optical emission-line luminosities with those of one million model spectral energy distributions. We explore the dependence of the resulting SFHs on galaxy stellar mass and redshift. We And that the average SFHs of high-mass galaxies rise and fall in a roughly symmetric bell-shaped manner, while those of low-mass galaxies rise progressively in time, consistent with the typically stronger activity of star formation in low-mass compared to high-mass galaxies. For galaxies of all masses, the star formation activity rises more rapidly at high than at low redshift. These findings imply that the standard approximation of exponentially declining SFHs widely used to interpret observed galaxy spectral energy distributions may not be appropriate to constrain the physical parameters of star-forming galaxies at intermediate redshifts.