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How to find discrete integral given different time intervals.

I want to implement a PID controller and I'm unsure of how to find the integration part. Normally the integral is calculated as $\sum_{n=1}^{t} e_{n}$, where $e_n$ is the sample error at time n. ...
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Is the integral in a PID controller definite or indefinite?

Would the integral calculated by the PID controller be considered definite or indefinite?