Why every integration rule ends in + C, with no differentiation needed. An integral measures what accumulated, not what the total is — C holds the place of the starting state, and adding it turns "how much arrived" into "how much is there".
Every integration rule ends in . The constant rule, the power rule, the trigonometric ones — all of them. It is usually presented as something to remember rather than something to understand, and there is a very simple reason for it.
An integral measures accumulation: how much arrived while you were watching. That is all the flow function can possibly tell it.
What it cannot tell you is how much was already there before you started. Nothing in the function carries that information, so the integral has to leave a space for it. That space is .
is the initial state of the system — its value before the accumulation began.
Both climb at the same rate, because the flow is the same. Only the starting height differs, and that height is .
Water flows in at litres per minute, and the tank starts empty.
Here the accumulated amount and the amount in the tank happen to be the same number, which is exactly why an empty tank hides the whole issue.
Same flow of litres per minute, but the tank already held litres when the clock started.
The integral part is identical in both examples, because the same water went in. Only changed, and with it the answer to what is actually in the tank.
The is not part of any particular rule. It is there because none of these rules were given a starting point. Supply one — the tank held litres, the car had already travelled kilometres — and becomes a definite number instead of a letter.