The real problem is solving the differential equation, for different values of parameters. To do this it's useful to write the solution in a useful basis of functions and using interpolation for everything else.

It would be better if you give an example that covers your problem more precisely. Can you share the code you used it for solving the problem?

Add an extra variable (call it s(t), for example) and add the differential equation

ds/dt = g(t,x)

with initial value s(t0) = 0. Then at the end time T, s(T) is the numerical estimate of the integral of g(t,x(t)) from t = t0 to t = T. It has the same order of accuracy as every other part of the solution to the differential equation.