## Tuesday, 3 December 2013

### Constant of Integration

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 $\cos(x)$:
\begin{align} \frac{d}{dx}[\sin(x) + C] &= \frac{d}{dx}[\sin(x)] + \frac{d}{dx}[C] \\ &= \cos(x) + 0 \\ &= \cos(x) \end{align}