The current becomes charge on the capacitor ... the capacitor until the output hits the power supply rail. Your integrator circuit then needs to work on a signal with no dc component.

To demonstrate the capability of loop probes, researchers at Universidad Carlos III and Università di Pisa developed an integrator circuit using two op amps and a passive RC filter that lets loop ...

The Design Idea in Figure 1 uses the S 1 switch of the Texas Instruments IVC102 precision integrator to select ... format proportional to the input-current ratio, realizing a true digital conversion.