Control Theory Integrator Phase Shift
The simple way (to me) is that with a differentiator, the rising positive slope of a sine wave input (as it passes through zero) produces a positive output peak. When the sine wave is falling, a negative peak is produced.

When integrating (accumulating), if the input sinwave is passing through zero, there is no change in the integrators output (nothing to accumulate). This corresponds with a positive or negative peak in the output and, if the input is rising through zero then the integrator's output (although flat when the sine input is exactly at zero) will begin to rise in a positive direction: -



A significant problem with regular op-amp integrators is that they also invert the signal and this leads to an extra confusion.