# Evaluating the Constant of Integration

## Key Questions

• If $G \left(x\right)$ is an antiderivative of $f \left(x\right)$ and $G \left({x}_{0}\right) = {y}_{0}$, then

$G \left(x\right) = \int f \left(x\right) \mathrm{dx} = F \left(x\right) + C$,

where $F \left(x\right)$ is any antiderivative of $f \left(x\right)$.

Since

$G \left({x}_{0}\right) = F \left({x}_{0}\right) + C = {y}_{0}$,

we have

$C = {y}_{0} - F \left({x}_{0}\right)$.

I hope that this was helpful.

• Normally, we want this integral function to be specified with a capital $f$, so that we can specify the antiderivative as $f \left(x\right)$.

However, using your variable naming, let's say that $F \left(x\right)$ is the antiderivative of $f ' \left(x\right)$, then by the Net Change Theorem, we have:

$f \left(x\right) = F \left(x\right) + C$

Therefore, the constant of integration is:

$C = f \left(x\right) - F \left(x\right)$
$= f \left(2\right) - F \left(2\right)$
$= 1 - F \left(2\right)$

This is a simple answer, however for many students, it is very difficult to this this abstractly. So, let's look at a concrete example:

$F \left(x\right) = {x}^{3}$ to match your variables
$F ' \left(x\right) = f ' \left(x\right) = 3 {x}^{2}$ to match your variables
$f \left(x\right) = \int 3 {x}^{2} \mathrm{dx}$
$= {x}^{3} + C$
$= F \left(x\right) + C$

Now, substitute the given values:

$f \left(2\right) = {x}^{3} + C = 1$
${2}^{3} + C = 1$
$F \left(2\right) + C = 1$
$C = 1 - F \left(2\right)$

So, if an abstract problem makes it difficult for you to find a solution, start with a concrete one to help you find a pattern.

• If $F \left(x\right)$ is an antiderivative of a function $f \left(x\right)$, that is,

$F ' \left(x\right) = f \left(x\right)$,

then

$G \left(x\right) = F \left(x\right) + C$, where $C$ is any constant,

is also an antiderivative of $f \left(x\right)$ since

$G ' \left(x\right) = \left[F \left(x\right) + C\right] ' = F ' \left(x\right) = f \left(x\right)$.

Hence, there are a family of functions (only differ by a constant) that are antiderivatives of $f \left(x\right)$. In order to include all antiderivatives of $f \left(x\right)$, the constant of integration $C$ is used for indefinite integrals.

$\int f \left(x\right) \mathrm{dx} = F \left(x\right) + C$

The importance of $C$ is that it allows us to express the general form of antiderivatives.

I hope that this was helpful.