Learning Outcomes
- Determine basic antiderivatives.
Section 4.3 Elementary antiderivatives (IN3)
Subsection 4.3.1 Activities
Definition 4.3.1.
If \(g\) and \(G\) are functions such that \(G' = g\text{,}\) we say that \(G\) is an antiderivative of \(g\text{.}\)
The collection of all antiderivatives of \(g\) is called the general antiderivative or indefinite integral, denoted by \(\int g(x)\,dx\text{.}\) All antiderivatives differ by a constant \(C\) (since \(\frac{d}{dx}[C]=0\)), so we may write:
\begin{equation*}
\int g(x)\,dx=G(x)+C\text{.}
\end{equation*}
Activity 4.3.2.
Consider the function \(f(x)=\cos x\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
- \(\displaystyle \sin x\)
- \(\displaystyle \cos x\)
- \(\displaystyle \tan x\)
- \(\displaystyle \sec x\)
Activity 4.3.3.
Consider the function \(f(x)=x^2\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
- \(\displaystyle 2x \)
- \(\displaystyle \frac{1}{3}x^3 \)
- \(\displaystyle x^3\)
- \(\displaystyle \frac{2}{3}x^3 \)
Remark 4.3.4.
We now note that whenever we know the derivative of a function, we have a function-derivative pair, so we also know the antiderivative of a function. For instance, in Activity 4.3.2 we could use our prior knowledge that
\begin{equation*}
\frac{d}{dx}[\sin(x)] = \cos(x)\text{,}
\end{equation*}
to determine that \(F(x) = \sin(x)\) is an antiderivative of \(f(x) = \cos(x)\text{.}\) \(F\) and \(f\) together form a function-derivative pair. Every elementary derivative rule leads us to such a pair, and thus to a known antiderivative.
In the following activity, we work to build a list of basic functions whose antiderivatives we already know.
Activity 4.3.5.
Use your knowledge of derivatives of basic functions to complete Table 86 of antiderivatives. For each entry, your task is to find a function \(F\) whose derivative is the given function \(f\text{.}\)
| given function, \(f(x)\) | antiderivative, \(F(x)\) |
| \(k\text{,}\) (\(k\) is constant) | |
| \(x^n\text{,}\) \(n \ne -1\) | |
| \(\frac{1}{x}\text{,}\) \(x \gt 0\) | |
| \(\sin(x)\) | |
| \(\cos(x)\) | |
| \(\sec(x) \tan(x)\) | |
| \(\csc(x) \cot(x)\) | |
| \(\sec^2 (x)\) | |
| \(\csc^2 (x)\) | |
| \(e^x\) | |
| \(a^x\) \((a \gt 1)\) | |
| \(\frac{1}{1+x^2}\) | |
| \(\frac{1}{\sqrt{1-x^2}}\) |
Activity 4.3.6.
Using this information, which of the following is an antiderivative for \(f(x) = 5\sin(x) - 4x^2\text{?}\)
- \(F(x) = -5\cos(x) +\frac{4}{3}x^3\text{.}\)
- \(F(x) = 5\cos(x) + \frac{4}{3}x^3\text{.}\)
- \(F(x) = -5\cos(x) - \frac{4}{3}x^3\text{.}\)
- \(F(x) = 5\cos(x) - \frac{4}{3}x^3\text{.}\)
Activity 4.3.7.
Find the general antiderivative for each function.
(a)
\begin{equation*}
f(x) = -4 \, \sec^2\left(x\right)
\end{equation*}
(b)
\begin{equation*}
f(x) = \frac{8}{\sqrt{x}}
\end{equation*}
Activity 4.3.8.
Find each indefinite integral.
(a)
\begin{equation*}
\int (-9 \, x^{4} - 7 \, x^{2} + 4) \, dx
\end{equation*}
(b)
\begin{equation*}
\int 3 \, e^{x}\, dx
\end{equation*}
