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We calculate the integrated galactic initial stellar mass function (IGIMF) inthe presence of IMF variations in clusters. IMF Variations for a population ofclusters are taken into account in the form of Gaussian distributions of theIMF parameters. For the tapered power law function used here, these are theslopes at the high and low mass ends, $\Gamma$ and $\gamma$, and thecharacteristic mass $M_{ch}$. Variations are modeled by varying the width ofthe Gaussian distributions. The reference values are the standard deviations ofthe parameters observed for young clusters in the present-day Milky Way$\sigma_{\Gamma}=0.6$, $\sigma_{\gamma}=0.25$, and $\sigma_{M_{ch}}=0.27$M$_{\odot}$. Increasing the dispersions of $\gamma$ and $\Gamma$ moderatelyflattens the IGIMF at the low and high mass ends. Increasing $\sigma_{M_{ch}}$shifts the peak of the IGIMF to lower masses, rendering the IGIMF more bottomheavy. This can explain the bottom heavy stellar mass function of Early-typegalaxies as they are the result of the merger of disk galaxies where thephysical conditions of the star forming gas vary significantly both in time andspace. The effect of IMF variations is compared to that due to other effectssuch as variations in the shape of the initial cluster mass function,metallicity, and galactic SFR. We find that the effect of IMF variations is adominant factor that always affects the characteristic mass of the IGIMF. Wecompare our results to a sample of ultra-faint dwarf satellite galaxies (UFDs).Their present-day stellar mass function is an analog to their IGIMF at the timetheir stellar populations have formed. We show that the slope of the IGIMF ofthe UFDs can only be reproduced when IMF variations of the same order as thosemeasured in the present-day Milky Way are included. (Abridged)

The galaxy-wide stellar initial mass function in the presence of cluster-to-cluster IMF variations