The author identified the cause of the integration-to-zero behavior - they've set the DC (0Hz) component to 0, so the signal must sum to 0. Any length-N signal

The author identified the cause of the integration-to-zero behavior - they've set the DC (0Hz) component to 0, so the signal must sum to 0. Any length-N signal can be thought of as one period of an N-periodic signal, so the FFT will be exact. It's not the first and last samples that need to be identical, it's the 1st sample needs to match the (N+1)th sample (1-indexed), which comes right after the last sample of the original finite signal. For example if you have the length-8 signal 12345678, the FFT is valid - you can think of it as a slice of the signal ...123456781234567812345678...