Dark matter (DM) in cosmic structures is expected to produce signals originating from its particle physics nature, among which the electromagnetic emission represents a relevant opportunity. One of the major candidates for DM are weak-scale particles, however no convincing signal of them has been observed so far. For this reason, alternative candidates are getting increasing attention, notably sub-GeV particles, which are the subject of our work. The challenge in indirect detection of sub-GeV DM is that there is scarcity of competitive experiments in the energy range between 1 MeV and hundreds of MeV, hence we need to find alternative ways to study DM candidates with mass in this energy window. In our work we focused on the X-ray band. In particular, we included in the total flux the contribution from Inverse-Compton scattering (ICS) of the low-energy photons in our galaxy, due to the electrons and positrons produced by annihilation events of DM particles. After the ICS, these radiation fields become X-rays with an energy covered by the INTEGRAL data, which we used to derive conservative bounds on the DM annihilation cross section. We obtained the strongest constraints at present time for DM particles with a mass between 150 MeV and 1.5 GeV.