We present a study of the infrared properties of X-ray selected, moderate luminosity (Lx=10^{42}-10^{44}ergs/s) active galactic nuclei (AGNs) up to z~3, to explore the links between star formation in galaxies and accretion onto their central black holes. We use 100um and 160um fluxes from GOODS-Herschel -the deepest survey yet undertaken by the Herschel telescope- and show that in >94 per cent of cases these fluxes are dominated by the host. We find no evidence of any correlation between the X-ray and infrared luminosities of moderate AGNs at any redshift, suggesting that star-formation is decoupled from nuclear (AGN) activity. The star formation rates of AGN hosts increase strongly with redshift; by a factor of 43 from z<0.1 to z=2-3 for AGNs with the same X-ray luminosities. This increase is consistent with the factor of 25-50 increase in the specific star formation rates (SSFRs) of normal, star-forming (main-sequence) galaxies. Indeed, the average SSFRs of AGN hosts are only marginally (20 per cent) lower than those of main-sequence galaxies, with this small deficit being due to a fraction of AGNs residing in quiescent (low-SSFR) galaxies. We estimate 79+/-10 per cent of moderate AGNs are hosted in main-sequence galaxies, 15+/-7 per cent in quiescent galaxies and <10 per cent in strongly starbursting galaxies. The fractions of all main sequence galaxies at z<2 experiencing a period of moderate nuclear activity is strongly dependent on galaxy stellar mass (Mstars); rising from a few per cent at Mstars~10^{10}Msun to >20 per cent at Mstars>10^{11}Msun. Our results indicate that it is galaxy stellar mass that is most important in dictating whether a galaxy hosts a moderate luminosity AGN. We argue that the majority of moderate nuclear activity is fuelled by internal mechanisms rather than violent mergers, suggesting that disk instabilities could be an important AGN feeding mechanism.
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