If you have a system which inherently possesses integrating characteristics (such as charge accumulation on a very low leakage node), you must reset it periodically for two reasons.
First, to establish the baseline at all: an electronic integrator performs the mathematical function of a definite integral. One cannot integrate without a point to start from, i.e., the integral of so-and-so is this-and-that plus a constant, a constant which is undefined without reference.
Second, because an input with an average offset will eventually cause the output to saturate and the integrator circuit ceases to be an integrator as such.
The normal approach is to arrange the system so that resetting is not required (e.g., by placing a sufficiently large feedback resistor to the integrator input, so that at very low frequencies, it tends to drift back to zero, thus turning it into a very high DC gain amplifier rather than a true integrator), or by adding circuitry to add or subtract a set amount from the integral term to keep it bounded (if impulses or discrete pulses are used, they can be counted with an up-down counter to extend the range digitally: see sigma-delta ADC).
Tim