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The highside OPAMP differentiates the input while the lowside acts as an integrates it.
Using Ohm's Law, and knowing the formula for capacitor current;
i = CdV/dt,
we can determine which configurations act as an integrator or a differentiator.
The output voltage is determined by the voltage drop of the element connected to it.
If it is a resistor, then by Ohm's Law;
V = iR, where i is the capacitor current, we know that the output is just the differential form current multiplied to some constant, which still results to a differential equation.
Hence, it is a differentiator.
If a capacitor is connected to the output instead, the voltage across the capacitor is used, which is the "inverse" of the capacitor current formula, whoch uses an integral.
Therefore, it is an Integrator.
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