Active Integrator Parallel Capacitor in Feedback
All capacitors can integrate charge. An arbitrary value of x Coulombs of charge on a 10 nF capacitor results in a much larger potential across the capacitor compared to a 1 uF capacitor. The relation is $$ V = \dfrac{q}{C} $$

The impact of this, is that, you are setting the effective integral gain of the amplifier by choice of the feedback integration capacitor.

An ideal integrator has a frequency response of -20 dB/dec. So, you need to decide what gain you wish to have at some frequency. Perhaps 0 dB at 10 kHz, resulting in 60 dB at 10 Hz.

To achieve an approximately ideal response from your integrator, you will want the effective feedback factor of C1, C2, and R1 to be at least 20 dB below the open-loop gain of the amplifier.  

There are also non-idealities of capacitors to watch out for: leakage, voltage-coefient, dielectric absorption, temp-co, etc... Your choice of capacitor should include these effects.