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 Î (a,b), then G is called an antiderivative of f.

We can construct antiderivatives by integrating. The function

 F(x) = óő 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 Î (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 ddx [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)
 Ű
 óő f(x) dx = F(x)+C.
 d dx [g(x)] = g˘(x)
 Ű
 óő g˘(x) dx = g(x)+C.

Properties of the Indefinite Integral

 ddx [ň f(x) dx] = f(x)

proof:

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

 d dx éë óő f(x) dx ůű
 =
 d dx [ F(x) ]
 =
 f(x).

• (Linearity) ň[af(x)+bg(x)] dx = aň f(x) dx+bň g(x) dx.

proof:

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

 d dx éë a óő f(x) dx+b óő g(x) dx ůű
 =
 a d dx éë óő f(x) dx ůű +b d dx éë óő g(x) dx ůű
 =
 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[ň x5 dx] = x5.

Key Concept

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

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