      Custom Search   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.   Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business