a slope of $\beta=1.5\pm 0.8$ (1$\sigma$ error), and thus $\alpha=0.5 \pm 0.8$, which is similar to the slope found in other spiral galaxies (e.g., $\alpha\sim0.4-0.5$; the Antennae galaxies, @2002ApJ...577..726Z; M82, @2007ApJ...661..135Z; M81, @2001ApJ...549L..43T). In these galaxies, the X-ray source population is mostly dominated by HMXBs. The ULX N4088–X1, with a luminosity $L_\mathrm{0.2-10.0 keV} = 3.4\ \times$ 10$^{39}$ erg s$^{-1}$, could thus be located at the high end of the HMXB distribution of NGC4088, which is consistent with the association of ULXs with young stellar populations (e.g., @2002ApJ...577..726Z; @2004ApJS..154..519S). However, the possibility that the ULX is a LMXB (e.g., @2012MNRAS.420.2969M; @2012ApJ...750..152S) cannot be ruled out.

To further study the star-formation rate (SFR) of NGC4088, we overplot the XLF of star-forming galaxies from [@2012MNRAS.419.2095M], fig. 4, black line, on the XLF of NGC4088. These authors normalize their XLF to 1 $M_{\odot}$ yr$^{-1}$. By rescaling this value to fit our data (see Fig. \[diffXLF\]), it is possible to obtain an estimate on the SFR for NGC 4088. This provides an estimated rate of 4.5 $M_{\odot}$ yr$^{-1}$, in good agreement with the range of SFRs 1.7–7.8 $M_{\odot}$ yr$^{-1}$ obtained by [@2006ApJ...643..173S] in the H$\alpha$, IR, and radio bands.

![Fit of the differential XLF inside 