Ok, the DC component will get filtered due to AC coupling.
What about the offset that will be produced inside the opAmp and will get intgrated with the result.
How will that get through the capacitor ?. If you amplify by AC coupling before the integrator the offset in the integrator will be a much smaller percentage of the signal than if you do not. DC ampilfication before the integrator will, of course, amplify the amplifiers offset, so there is no advantage there.

Erik