If you change C, you change the function.

and

may have the same derivative, but for all x, the two functions are completely different. Your constant of integration is just that; a constant. Constants in no way affect your derivative and so your +C can be anything from

to

.

We don't always take the integral from b to a. That's only when we're finding accumulated area, in which case the +C is negligable. But sometimes we just want F(a) or F(b), in which case it's very important to know.

Say we know the antiderivative of

is

. And say we know that

. This gives us a point on the function with which to calculate C. We know that

, so C must equal 1.