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Antiderivatives

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

We can construct antiderivatives by integrating. The function

F(x) =
x

a 
f(t) dt
is an antiderivative for f since it can be shown that F(x) constructed in this way is continuous on [a,b] and F(x) = f(x) for all x (a,b).

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).

The process of finding antiderivatives is called antidifferentiation or integration:
d
dx
[F(x)] = f(x)
      
      
f(x) dx = F(x)+C.
d
dx
[g(x)] = g(x)
      
      
g(x) dx = g(x)+C.

Properties of the Indefinite Integral

d
dx
[ f(x) dx] = f(x)

    proof:

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

    d
    dx


    f(x) dx
    =
    d
    dx
    [ F(x) ]
    =
    f(x).

  • (Linearity) [af(x)+bg(x)] dx = a f(x) dx+b g(x) dx.

    proof:

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

    d
    dx

    a
    f(x) dx+b
    g(x) dx
    =
    a d
    dx


    f(x) dx
    +b d
    dx


    g(x) dx
    =
    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[ x5 dx] = x5.


Key Concept

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

We can construct antiderivatives by integrating. The function

F(x) =
x

a 
f(t) dt
is an antiderivative for f. In fact, every antiderivative of f(x) can be written in the form F(x)+C, for some C.

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