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A general OpAmp integrator with an R & C (resistor and capacitor) will do. The question will be how long you want to integrate (or accumulate) the signal. If your signal voltage level is in milliVolts then it will be easier to find a capacitor with required amount of capacitance and you can...
Probably the MOSFET is heating up and the conductivity is going down (even I don't know how the conductivity is going down). Check the temperature of the MOSFET
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