• We will consider integral calculus.
• We will study fundamental ideas such as areas under curves and antiderivatives.

In This Module

• We will look at the idea of an antiderivative.

The power rule tells us how to differentiate powers of \(x\).

For example

The functions \(x^2\) and \(2x\) are connected by the process of differentiation.

We call \(2x\) the derivative of \(x^2\).

We call \(x^2\) an antiderivative of \(2x\).

Finding an antiderivative of a function can be thought of as “undoing” differentiation.

In general, we ask:

When finding the antiderivative of \(f(x)=2x\), we are asking: “What function did we differentiate to get \(2x\)?”

We call \(F\) an antiderivative of \(f\).

While any function, \(f\), has only one derivative,

\(f\) will have more than one antiderivative.

In fact, if we can find one antiderivative of \(f\), then we can find infinitely many!

Check that \(\dfrac{1}{3}x^3\) and \(\dfrac{1}{3}x^3+2\) are both antiderivatives of \(f(x)=x^2\).

Solution

Using the power rule, we get

\(\dfrac{d}{dx}\left ( \dfrac{1}{3}x^3\right )\) \(=\dfrac{1}{3}\left( \dfrac{d}{dx}\Big (x^3 \Big)\right)\) \(=\dfrac{1}{3} \left ( 3x^2\right) = x^2\)

and

\(\dfrac{d}{dx}\left ( \dfrac{1}{3}x^3+2\right )\) \(= \dfrac{1}{3}\left( \dfrac{d}{dx}\Big (x^3 \Big) \right)+\dfrac{d}{dx}\Big(2\Big) \) \(=\dfrac{1}{3} \left ( 3x^2\right) + 0 \) \( = x^2\)

Notice that we could replace \(2\) with any constant, \(c\), and the final answer would not change.

By changing the constant, we can come up with infinitely many antiderivatives.

Consider the function \(f(x)=x^5\). Find an antiderivative of \(f\).

Solution

From the power rule

\(\dfrac{d}{dx}\Big ( x^6\Big )\) \(=6x^5\)

and so we include the appropriate constant \(\frac{1}{6}\) to get

and so, \(F(x)=\dfrac{1}{6}x^6\) is an antiderivative of \(f\).

Find an antiderivative of \(f(x)=2x^3\).

Solution

By the power rule, we consider a function of the form \(x^4\).

and so, with a small adjustment, we get

\(\dfrac{d}{dx}\left ( \dfrac{1}{2}x^4\right )\) \(=\dfrac{1}{2}\left ( 4x^3\right )\) \(=2x^3\)

Therefore, \(F(x)=\dfrac{1}{2}x^4\) is an antiderivative of \(f(x)=2x^3\).

Note that, while it takes some effort to find an antiderivative of \(f\), it is easy to check if your answer is correct!