Since there is a great deal of integration of rotating complex exponentials, and rotating complex exponentials are composed of sinusoids, it's a good idea to look at what happens if we integrate a sinusoid.
If we integrate a sinusoid over a single period, the result is exactly zero. The positive and negative areas cancel out.
If we multiply two sinusoids of the the same frequency and the same phase, we get a positive result. If the phase is shifted by 180 degrees, we get a negative result.
