Would the integral calculated by the PID controller be considered definite or indefinite?
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According to the definition given in this WP article, the relevant integral is of the form
If this is regarded as a function of $t$ (say in a theoretical treatment), it would be an indefinite integral; however, your question concerns the integral calculated by the PID controller, and this is a definite integral for each specified value of $t$.
Depends on the control circuit but hardly ever a precise integration.
If the controller is realized as a RLC circuit (an analog controller) then maybe you might say it is a definite integral but they most often come with a reset (anti-windup) algorithm to avoid saturation and kickback hence the integral is taken from the reset time to the actual time.
On the other hand if it is a discrete time controller, you first filter it out so $e(t)$ gets convoluted with a low-pass filter and then due to saturation problems you might wish to make the $I$ parameter gain dependent. If the error is too large you decrease the $I$ parameter of the controller etc.
Therefore, theoretically it would be a definite integral but to avoid oscillations around the reference signal and for other practical matters it is almost never implemented as a pure integrator.