Constant of Integration - Origin of The Constant

Origin of The Constant

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.

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:

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 :

\begin{align} \frac{d}{dx} &= \frac{d}{dx} + \frac{d}{dx} \\ &= \cos(x) + 0 \\ &= \cos(x) \end{align}

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