page 1 of 2

Page 1

Page 2

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

Page 1

Page 2

Take a Study Break!