RC Integrator circuit final voltage
That's not an integrator- it's a low pass filter. Just as well, as a true integrator has an arbitrary constant in the output so you would not be able to predict the steady-state voltage (in a real analog integrator it would tend to drift towards one rail or the other due to offset even if the input averaged zero). 

Suggest you assume a starting output voltage V0 at the beginning of a cycle and show that it's in steady state (returns to the same voltage for the next cycle) rather than trying to model the transient start up condition.