[Buraksr]

What I am meaning is the input to the transistor on a chip is highly volatile, so the aproxmation of its power consumption using infinite or long periods of time of a single value is misleading.

If a transistor is only reciving 5v than it is fair to say the power consumed is equal.

But realisticly the input could change every clock cycle and we would see the cases vary.

The reason I only integrate to tao was to show if the input switches before t=infinity then a diffrence between cases clearly manifest. But as you say, when it does not switch for time t>>tao the diffrence is small.