You seem to be trying to see things that may not be observable.
1) There is no such thing as "the integral of a line". Remember that arbitrary constant? Which is it that you think should mean something to you?
2) Constant terms vanish under differentiation. This might give some insight. If you hae not already encountered it, you will find it useful in simple motion problems with uniform acceleration.
2a) x(t) might be the displacement of some object at time t.
2b) x'(t) = v(t) might be the velocity of some object at time t.
2c) x"(t) = v'(t) = a(t) might be the acceleration of some object at time t.