antiderivative
An antiderivative answers the question, what function had this as its derivative? If a derivative tells you a car's speed at every instant, an antiderivative reconstructs its position. Reversing differentiation is the central computational trick behind evaluating integrals via the fundamental theorem.
Formally, F is an antiderivative of f on an interval I if F is differentiable on I and F'(x) = f(x) for all x in I. Antiderivatives are never unique: if F is one, so is F + C for any constant C, and on a connected interval these are the only ones, since two functions with equal derivatives differ by a constant (a consequence of the mean value theorem).
Having an antiderivative and being integrable are different properties. A function can possess an antiderivative without being continuous — Darboux's theorem says any derivative has the intermediate-value property, so derivatives can be quite wild yet still be antiderivatives. Conversely, an integrable function may fail to have any antiderivative; for instance a step function with a jump cannot be the derivative of anything, because derivatives never jump.
An antiderivative of f(x) = cos x is F(x) = sin x, since (sin x)' = cos x; the general antiderivative is sin x + C. Checking: differentiating sin x + C recovers cos x for every constant C.
Antiderivatives come in a one-parameter family differing by a constant.
The symbol for the indefinite integral denotes the family of antiderivatives, which is why a constant of integration is always appended. The definite integral, by contrast, is a single number; the link between them is exactly the fundamental theorem.