numerical integration of the ERROR in a PI controller

pid3control

good day all.i've just written a program in fortran77 for the model of a binary distillation column.i included a PI controller in my program.To obtain the integral term of the PI,I used the explicit euler to numerically integrate the error.i would love to compare my results using another method to numerically integrate the error.but i'm confused.the euler method was quite simple.i defined ERINT as the integral of the error (∫E).and since E(i.e the error) is the derivative of the integral i simply employed euler to solved the resulting ODE.That is,
ERINT(n+1)=ERINT(n) + TE(n) where T is the step size
please which other method simple can i use to obtain the integral term.can i also see simple chemical processes (in fortran77) where they are applied in a PI controller it obtain the integral term? please an answer would be greatly appreciated.many thanks