INDEFINITE INTEGRALS

When we were finding the derivative of a function, we wrote

where the derivative of F(x) is f(x). Our problem is to find F(x) when we are given f(x).

We know that the symbol ... dx is the inverse of or when dealing with differentials, the operator symbols d and are the inverse of each other; that is,

and when the derivative of each side is taken, d annulling , we have

or where annuals , we have

From this, we find that

so that,

Also we find that

so that,

Again, we find that

so that,

This is to say that

and

where C is any constant of integration.

A number that is independent of the variable of integration is called a constant of integration. Since C may have infinitely many values, then a differential expression may have infinitely many integrals differing only by the constant. This is to say that two integrals of the same function may differ by the constant of integration. We assume the differential expression has at least one integral. Because the integral contains C and C is indefinite, we call

an indefinite integral of f(x) dx. In the general form we say

With regard to the constant of integration, a theorem and its converse are stated as follows:

Theorem 1. If two functions differ by a constant, they have the same derivative.

Theorem 2. If two functions have the same derivative, their difference is a constant.