Every indefinite integral ends in + C, and a definite one never does. C stands for the starting state, and a definite integral asks only for the change between two points — so the same C sits in both terms and subtracts itself away.
Every indefinite integral ends in . Put limits and on the sign and it is suddenly gone. Nothing was dropped and no rule was bent — simply stops mattering, and it is worth seeing exactly why.
records the starting state of the system — where the accumulation began before any of the integrating happened.
In the tank picture, is how much water was already sitting in the tank when you started the clock. An empty tank gives . A tank that was half full gives some larger . The flow going in afterwards is identical either way.
This is not a special case or a convention. It happens for every constant, every function and every pair of limits, which is why a definite integral never carries a .
The rise between and is therefore untouched. A definite integral asks for that rise, and the rise does not care where the line started.
Water flows in at litres per minute. How much collects between minute and minute ?
Two different tanks, two different totals in them, one identical answer. The question was never how much water is in the tank — it was how much arrived during those two minutes.
Evaluate , keeping in place the whole way.
The terms subtract themselves away on the second-to-last line. Leaving out of a definite integral is not a shortcut — it is what the algebra does on its own.