Indefinite Integral

In computing integrals of f, we usually work with the family of all anti-derivatives of f. We shall call this whole family of functions the indefinite integral of f. The symbol for the indefinite integral is f(x)dx. If F(x) is one antiderivative of f, the indefinite integral is the set of all functions of the form F(x) + C_o, C_o constant. We express this with the equation

f (x) dx = F(x) + C.

It is an equation between two families of functions rather than between two single functions. C is called the constant of integration. To illustrate the notation,

3x² dx = x³ + C.

We repeat the above definitions in concise form.

When working with indefinite integrals, it is convenient to use differentials and dependent variables. If we introduce the dependent variable u by u = F(x), then

du = F'(x) dx = f(x) dx.

Thus the equation

f (x) dx = F(x) + C

can be written in the form 

du = u + C.

The differential symbol d and the indefinite integral symbol behave as inverses to each other. We can start with the family of functions u + C, form du, and then form du = u + C to get back where we started. Some of the rules for differentiation given in Chapter 2 can be turned around to give a set of rules for indefinite integration.