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We characterize the incidence of active galactic nuclei (AGNs) is 0.3 < z < 1 star-forming galaxies by applying multi-wavelength AGN diagnostics (X-ray, optical, mid-infrared, radio) to a sample of galaxies selected at 70-micron from the Far-Infrared Deep Extragalactic Legacy survey (FIDEL). Given the depth of FIDEL, we detect "normal" galaxies on the specific star formation rate (sSFR) sequence as well as starbursting systems with elevated sSFR. We find an overall high occurrence of AGN of 37+/-3%, more than twice as high as in previous studies of galaxies with comparable infrared luminosities and redshifts but in good agreement with the AGN fraction of nearby (0.05 < z < 0.1) galaxies of similar infrared luminosities. The more complete census of AGNs comes from using the recently developed Mass-Excitation (MEx) diagnostic diagram. This optical diagnostic is also sensitive to X-ray weak AGNs and X-ray absorbed AGNs, and reveals that absorbed active nuclei reside almost exclusively in infrared-luminous hosts. The fraction of galaxies hosting an AGN appears to be independent of sSFR and remains elevated both on the sSFR sequence and above. In contrast, the fraction of AGNs that are X-ray absorbed increases substantially with increasing sSFR, possibly due to an increased gas fraction and/or gas density in the host galaxies.