2004 Cadillac SRX revers lights

The rear integration module grounds the control side of the reverse lamp relay, in response to a bus message that reverse is engaged. If the reverse lamp fuse is good, have you checked that it has power across it. Can you hear the relay click when reverse is selected, the relay may be serviceable, or may be integral to the fuse block. If there is power across the fuse, and you can or can not hear the relay click, swap the relay with one of similar type, there should be others in any of the fuse blocks. If the relay clicks, and swapping it had no resolve, you will need to check circuits from the relay to the rear extension harness connector, or even the reverse lamps. As far as what it could be, the relay, the rear integration module, any of the harness connections or circuitry, are possibilities.