Arbitrary constant of integration: Encyclopedia II - Arbitrary constant of integration - Where does the constant come from?

Arbitrary constant of integration - Where does the constant come from?

The derivative of any constant function is zero. Once one has found one antiderivative F, adding or subtracting a constant C will give us another antiderivative, because (F + C)' = F' + C' = F' . The constant is a way of expressing that every function has an infinite number of different antiderivatives.

For example, suppose one wants to find antiderivatives of cos(x). One such antiderivative is sin(x). Another one is sin(x)+1. A third is sin(x)-π. Each of these has derivative cos(x), so they are all antiderivatives of cos(x).

It turns out that adding and subtracting constants is the only flexibility we have in finding different antiderivatives of the same function. That is, all antiderivatives are the same up to a constant. To express this fact for cos(x), we write:

Replacing C by a number will produce an antiderivative. By writing C instead of a number, however, a compact description of all the possible antiderivatives of cos(x) is obtained. C is called the constant of integration. It is easily determined that all of these functions are indeed antiderivatives of cos(x):

Adapted from the Wikipedia article "Where does the constant come from?", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki