Constant of Integration

 Definite integral example

The derivative of any constant function is zero. Once one has found one antiderivative F(x) for a function f(x), adding or subtracting any constant C will give us another antiderivative, because (F(x) + C)' = F\,'(x) + C\,' = F\,'(x). 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 \cos(x). One such antiderivative is \sin(x). Another one is \sin(x)+1. A third is \sin(x)-\pi. 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:
\int \cos(x)\,dx = \sin(x) + C.
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):
\frac{d}{dx}[\sin(x) + C] &= \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[C] \\
                          &= \cos(x) + 0 \\
                          &= \cos(x)


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