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
Also we find that
Again, we find that
This is to say that
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.