Let f(x) be continuous on [a,b]. If G(x) is continuous on [a,b] and G¢(x) = f(x) for all x Î (a,b), then G is called an antiderivative of f.

We can construct antiderivatives by integrating. The function

Properties

Let F(x) be any antiderivative for f(x).

• For any constant C, F(x)+C is an antiderivative for f(x).

  proof:

  Since

  d dx

  [F(x)] = f(x),

  d dx [ F(x)+C ] = d dx [ F(x) ]+ d dx [C] = f(x)+0 = f(x)

  so F(x)+C is an antiderivative for f(x).

• Every antiderivative of f(x) can be written in the form

  F(x)+C

  for some C. That is, every two antiderivatives of f differ by at most a constant.

  proof:

  Let F(x) and G(x) be antiderivatives of f(x). Then F¢(x) = G¢(x) = f(x), so F(x) and G(x) differ by at most a constant (this requires proof---it is shown in most calculus texts and is a consequence of the Mean Value Theorem).

Properties of the Indefinite Integral

d dx

[ò f(x) dx] = f(x)

proof:

Let ò f(x) dx = F(x), where F(x) is an antiderivative of f. Then

d dx é ë ó õ f(x) dx ù û = d dx [ F(x) ] = f(x).

• (Linearity) ò[af(x)+bg(x)] dx = aò f(x) dx+bò g(x) dx.

  proof:

  We need only show that aò f(x) dx+bòg(x) dx is an antiderivative of ò [af(x)+bg(x)] dx:

  d dx é ë a ó õ f(x) dx+b ó õ g(x) dx ù û = a d dx é ë ó õ f(x) dx ù û +b d dx é ë ó õ g(x) dx ù û = af(x)+bg(x).

Examples

1. Every antiderivative of x² has the form x³/ 3 + C, since d/dx[x³/ 3] = x².
2. d/dx[ò x⁵ dx] = x⁵.

If G(x) is continuous on [a,b] and G¢(x) = f(x) for all x Î (a,b), then G is called an antiderivative of f.

We can construct antiderivatives by integrating. The function