Integration seen in one picture: the area piling up under a function is the height of its integral function. Move the stopping point and both numbers move together; a taller f delivers more area per step, so F climbs more steeply; and the area between a and b is F(b) - F(a), the subtraction in which the constant C cancels.
Integration is usually taught as a set of rules for turning one expression into another. The idea underneath is far simpler, and it can be seen in a single picture: the area piling up under a function is the height of its integral function.
Take the constant function . Its integral function is , and for now we can set the constant aside and use .
Now read the lower graph at the same place: . The area above and the height below are the same number, and they stay the same number wherever you stop.
Same two graphs, three different stopping points.
Nothing here depends on the numbers being easy. The rule is what defines the integral function in the first place.
The constant case hides something, because every step to the right adds the same slab of area. Take instead, where each step adds more area than the last.
More area arriving per step means has to climb faster to keep matching it. That is why the lower curve bends upward: its slope at each point is exactly the height of there.
If records the area accumulated up to a point, then the area between and is what has accumulated by , minus what had already accumulated by .
The upper limit is evaluated first and the lower limit is subtracted from it. The constant never survives this subtraction, which is why it does not appear in a definite integral.
The integral function is .
The region is a trapezium with parallel sides and and width , so its area is . The geometry and the integral agree, as they must.