  The stellar initial mass function (IMF) integrated over an entire galaxy is
an integral over all separate star-formation events. Since most stars form in
star clusters with different masses the integrated IMF becomes an integral of
the (universal or invariant) canonical stellar IMF over the star-cluster mass
function. This integrated IMF is steeper (contains fewer massive stars per
G-type star) than the canonical stellar IMF. Furthermore, observations indicate
a relation between the star-formation rate of a galaxy and the most luminous
stellar cluster in it. This empirical relation can be transformed into one
between the star-formation rate of a galaxy and a maximum cluster mass. 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
integrated IMF for massive stars. This integrated IMF varies with the
star-formation history of a galaxy. Notably, large variations of the integrated
IMF are evident for dwarf galaxies. One important result is that the number of
type II supernovae per star is supressed relative to that expected for a
canonical IMF, and that dwarf galaxies have a supressed number of supernovae
per star relative to massive galaxies. For dwarf galaxies the number of
supernovae per star also varies substantially depending on the galaxy assembly
history.
