well why don't you use a discrete integrating method, which is basically a sum

thorically the formula is
I(T*Dt)=I((T-1)DT)+i(t)*Dt
of course the smaller DT is the better the results are

if you use signed int and sufficient data size you could have good enough result. This would spare an op amp

(HINT = just use I(T) = I(T-1)+i(T) , you ll divie the thing when you'll need the result and you normally will gain precision, as there is no error in a integer addition, and there could be much in integer division i.e. 1/2=1 or 0)