Don't know what year your Integra is, but if it is back around the 1990 model, (as my Honda Accord EX was) the owners manual called for 10W-30 engine oil.
That being said, my Accord EX was NOT a good shifter. Would always grind going into 2nd gear, and my wife could never get it down into 1st without it grinding. This was the days before I understood anything about MTF's etc, if there even was such a thing. I even went so far as to change the transmission oil to a Motorcycle oil, (Golden Spectro 10W-40) but it made no difference, the Accord just would not shift good.
BTW, this 1990 Accord EX was the very worse modern day car I ever owned. The only thing that did not go wrong with it was the engine (thank god). (Made in Marysville Ohio) Kept it 18 months and traded for a 1992 Camry (one of the few that was made in Japan) which never gave me an iota of problems in 96,000 miles.
Speaking of smooth shifting manual transmissions, whenever I go to Europe, I lease a Renault Laguna, and they always shift great, like shifting through butter. If I ever decide to buy another manual transmission car, I will probably go with a Nissan as they share a lot of parts and technology with Renault. Both have the same CEO. (Before anyone comes up with any Renault bad raps, I've leased six of them since 1989 and never had any problems with them).