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: