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The integrator calculates the area under the input signal. At any point in time the capacitor charge current equals the input voltage of the signal divide by R. The capacitor "accumulate" all these values
Any first order RC lowpass combination (active or passive) can act as an integrator if the operating frequencies are large if compared with the lowpass corner frequency (at least a factor of 10 as a rule of thumb).
WHY? Because for high frequencies the phase shift approaches 90 deg. (ideal integrator: exact 90 deg.).