Antiderivatives

What we have learned so far is that a function $f(x)$ can represent any quantity and its derivative <Sign in to see all the formulas> represents the rate of change of that quantity with respect to something else. For example, $x$ could represent time, $f(x)$ could represent the distance from some point and <Sign in to see all the formulas> will represent speed. But in the real world, we cannot always collect information on $f(x)$ and then find the derivative. Experimental results often give <Sign in to see all the formulas> and not $f(x)$. In this situation we would like to have a way to get some information about $f(x)$ given that we know <Sign in to see all the formulas>. This process of finding information about $f(x)$ given that we know <Sign in to see all the formulas> is called antidifferentiation

Here is our formal definition of an antiderivative:

Antiderivative
A function $F(x)$ is an antiderivative of $f(x)$ on the closed interval $[a,b]$ if and only if $F(x)$ is continuous on $[a,b]$ and <Sign in to see all the formulas> for every $x$ in the open interval $(a,b)$.

It is important to note that an antiderivative is not unique. Recall that the derivative of a constant is zero and that the derivative of two differentiable functions is the sum of their derivatives. Let's say that $F(x)$ is an antiderivative of $f(x)$. We can add a constant $C$ to $F(x)$ to get a new antiderivative $G(x)$. In other words, <Sign in to see all the formulas>. To see that $G(x)$ is also an antiderivative do the following:

<Sign in to see all the formulas>

All that is left is to show that $G(x)$ is continuous on $[a,b]$. The complete proof can be found here. Finally, you will oftentimes find an antiderivative written with a "+$C$" on the end to denote that it is not unique and can differ by a constant.

Let's end the lesson with a quick example. Find an antiderivative of <Sign in to see all the formulas>. We are looking for a function $F(x)$ where <Sign in to see all the formulas>. Now we know that

<Sign in to see all the formulas>

If <Sign in to see all the formulas>, then <Sign in to see all the formulas>. Now we need to show that $F(x)$ is continuous. Since $F(x)$ is a polynomial we know it is continuous. Therefore, <Sign in to see all the formulas> is an antiderivative of <Sign in to see all the formulas>. Finally, as mentioned above this will oftentimes be written as <Sign in to see all the formulas>.

Finally, this problem gives the general formula for finding the antiderivative of a polynomial. It should be memorized as it will used a lot later when solving for integrals.