In this section we present the fundamental theorem of calculus. First, we must make a definition. A function F(x) is called an antiderivative of a function f (x) if f (x) is the derivative of F(x) ; that is, if F'(x) = f (x) . The antiderivative of a function f (x) is not unique, since adding a constant to a function does not change the value of its derivative:
|(f (x) + c) = (f (x)) + (c) =|
It is true, however, that any two antiderivatives of a function f (x) will differ by a constant.
Why are antiderivatives a useful concept? Because they provide a shortcut for calculating definite integrals, as shown by the first part of the fundamental theorem of calculus.
Let f (x) be a function that is integrable on the interval [a, b] and let F(x) be an antiderivative of f (x) (that is, F'(x) = f (x) ). Then
|f (x)dx = F(b) - F(a)|
Since the expression F(b) - F(a) is one we will encounter often, we will sometimes employ a special shorthand to simplify our equations:
|F(x)|a b = F(b) - F(a)|
Note that any antiderivative F(x) will give the same value for F(b) - F(a) , since any two antiderivatives differ only by a constant, which will cancel upon subtraction. The formula in the theorem may also be written