It isn’t the head gasket. That would cause operating problems.
Odds are excellent that it’s the valvecover gasket.
Valvecover gaskets prevent seepage of oil past the junction between the head’s surface and the valvecover. When the engine is operating, the area under the valvecover becomes pressurized by “blowby”, gasses that get blown by the combustion pressures past the piston rings and pressurize the crankcase. The crankcase is openly connected to the space under the valvecover by the “return paths” through which oil that was used under pressure to lubricate the camshafts runs back to the crankcase. When an engine gets old, the rubber valvecover gaskets become compressed and lose their ability to properly seal the aforementioned periphery around the bottom of the valve cover. Pressure from the crankcase then causes oil to seep past the valvecover gasket. This is very common, and because rubber gaskets constantly under compression change their ability to seal over time, and because crankcases can become more pressurized as “blowby” increases due to normal wear, this seepage is extremely common and pretty much inevitable.
Now, the good news. I’ve never done an '87 Integra, but valvecover gaskets are inexpensive to buy and generally removable/replaceable without major effort such as engine removal. If there were engine options, tell us which you have and perhaps one of the regulars here can tell you exactly what’s involved in the gasket replacement.
More goo0d news; as long as you don’t let the oil level run low, seepage past the gasket won’t cause other damage… but if it’s burning on the exhaust manifold, the acrid smoke isn’t pleasant… or healthy… to breath.