Integrator as DC servo
Yes, it will affect the AC signal by a small amount and that amount halves for every doubling of the AC frequency so, you design an integrator that is intended to remove DC so as to not cause a significant effect on the amplifier’s AC processing gain.

Look at R5 and C3; they form a low pass filter at about 0.2 Hz so, at 0.4 Hz the integrating effect is halved and at 0.8 Hz the effect has quartered. Do you follow where I am going with this?