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The integrated galaxial initial mass function (IGIMF) is the relevant distribution function containing the information on the distribution of stellar remnants, the number of supernovae and the chemical enrichment history of a galaxy. Since most stars form in embedded star clusters with different masses the IGIMF becomes an integral of the assumed (universal or invariant) stellar IMF over the embedded star-cluster mass function (ECMF). For a range of reasonable assumptions about the IMF and the ECMF we find the IGIMF to be steeper (containing fewer massive stars per star) than the stellar IMF, but below a few Msol it is invariant and identical to the stellar IMF for all galaxies. However, the steepening sensitively depends on the form of the ECMF in the low-mass regime. Furthermore, observations indicate a relation between the star formation rate of a galaxy and the most massive young stellar cluster in it. The assumption that this cluster mass marks the upper end of a young-cluster mass function leads to a connection of the star formation rate and the slope of the IGIMF above a few Msol. The IGIMF varies with the star formation history of a galaxy. Notably, large variations of the IGIMF are evident for dE, dIrr and LSB galaxies with a small to modest stellar mass. We find that for any galaxy the number of supernovae per star (NSNS) is suppressed relative to that expected for a Salpeter IMF. Dwarf galaxies have a smaller NSNS compared to massive galaxies. For dwarf galaxies the NSNS varies substantially depending on the galaxy assembly history and the assumptions made about the low-mass end of the ECMF. The findings presented here may be of some consequence for the cosmological evolution of the number of supernovae per low-mass star and the chemical enrichment of galaxies of different mass.