just google pitot-tube and read up on what it does.
afaik modern engines adjust fuel flow to match a certain EPR (Engine Pressure Ratio - measured as a difference of air flowing in and out of the engine). of course they also have limits regarding temperatures and rotational speeds.
and why would you integrate pressures? you mean subtract perhaps.
so you measure temperature and some form of airspeed and you get a mach-number? i'm quite sure some good reading material is out there including here on PPRuNe if you just use google with the keywords you used in your question.
anyhow why would it need mach-number, i don't know that it does, how did you get to that conclusion?