### Constant of Integration

The derivative of any constant function is zero. Once one has found one antiderivative for a function , adding or subtracting any constant

*C*will give us another antiderivative, because . The constant is a way of expressing that every function has an infinite number of different antiderivatives.

In other words in finding the indefinite integral of a function f(x) , an arbitrary constant is to be added to the result to make it general . This is the reason why the integral is referred to as an indefinite integral . The arbitrary constant is usually referred to as the constant of integration .

It is easily seen , however , that in evaluating a definite integral this constant of integration cancels out and its value is thus definite .

For the shake of convenience the arbitrary constant of integration has generally been omitted but it is always understood to be present in every case , and should be supplied by the students in the result .

For example, suppose one wants to find antiderivatives of . One such antiderivative is . Another one is . A third is . Each of these has derivative , so they are all antiderivatives of .

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:

*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 :

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