Since no one has officially said how to actually replace this, I'll give my input.
I currently have a Solara that the lights turn back on after the dome light timer times out.
The integration relay is a white plastic box on the back of the fuse block.
It does not require removing the fuse block to get it off.
It is a white plastic box approx 3" x 5" and approx 1.5" thick.
To remove it turn yourself upside down under the dash, and look up behind the fuse block.
You will see two connectors one which is large with approx 10 wires in it, and a small one with 4 or so.
Push in the locking tab, and remove the two connectors.
Now just slightly to the right you will see a locking tab that latches the integration relay box to the back of the fuse block.
Pull up the tab to loose the bottom from the fuse block.
Now either by pulling and/or prying with a large screw driver, the entire integration block pops out of the fuse block toward the front of the vehicle. Just like unplugging a big giant relay. Be careful not to apply too much upward or angled stress on the assembly. The direction is out away from the fuse block.
Inside is a very complex circuit board. I seriously doubt unless you can find a full blown schematic you will be able to repair this. There are multiple relays, an embedded logic/timer micro, many surface mount diodes, and quite a few output transistors.
I'm obviously going to have to find a replacement.
Not sure if these are all basically the same, and I can get one from a bone yard, or if I have to buy a new one.
I have to believe that this is not a cheep item.
Hope this helps.