An antiderivative of a function f is a function whose derivative is f. In other words, F is an antiderivative of f if F' = f. To find an antiderivative for a function f, we can often reverse the process of differentiation.
For example, if f = x^{4}, then an antiderivative of f is F = x^{5}, which can be found by reversing the power rule. Notice that not only is x^{5} an antiderivative of f, but so are x^{5} + 4, x^{5} + 6, etc. In fact, adding or subtracting any constant would be acceptable.
This should make sense algebraically, since the process of taking the derivative (i.e. going from F to f) eliminates the constant term of F.
Because a single continuous function has infinitely many antiderivatives, we do not refer to "the antiderivative", but rather, a "family" of antiderivatives, each of which differs by a constant. So, if F is an antiderivative of f, then G = F + c is also an antiderivative of f, and F and G are in the same family of antiderivatives.
The notation used to refer to antiderivatives is the indefinite integral. f (x)dx means the antiderivative of f with respect to x. If F is an antiderivative of f, we can write f (x)dx = F + c. In this context, c is called the constant of integration.
To find antiderivatives of basic functions, the following rules can be used: