What year is your car?

'94-'97 Integras had a mechanical latch, meaning it will open by the cable or key. All '98+ years worked on a circuit. When you pull the lever inside, it completes the circiut and pops the trunk open, same thing with using the key.

So without knowing what year the car is, I can't really help with diagnosing the problem as it could be from a popped fuse to a broken cable clip.