Our cars have on-board diagnostics, you just have to pull the passenger side foot board and jump the green connector service port wire with a paper clip to bring up the codes. Count the flashes then look up the code.

I bet it is your Main Fuel Relay if your CEL stays on when you try to start it. When your CEL comes on and stays on, then your fuel pump is not pressurizing the system. The Main Fuel Relay is notorious for going bad on 2nd gen integras and is an easy fix located behind the drivers side kick panel under the steering wheel. Hope this helps and you figure your problem out.