Skip to main content
Logo image

Preview of Team-Based Inquiry Learning Library

TBIL Community

Section 12.3 Elementary Antiderivatives (IN3)

Subsection 12.3.1 Activities

Definition 12.20.

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 12.21.

Consider the function \(f(x)=\cos x\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
  1. \(\displaystyle \sin x\)
  2. \(\displaystyle \cos x\)
  3. \(\displaystyle \tan x\)
  4. \(\displaystyle \sec x\)
Answer.
A. \(\sin x \)

Activity 12.22.

Consider the function \(f(x)=x^2\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
  1. \(\displaystyle 2x \)
  2. \(\displaystyle \frac{1}{3}x^3 \)
  3. \(\displaystyle x^3\)
  4. \(\displaystyle \frac{2}{3}x^3 \)
Answer.
B. \(\frac{1}{3} x^3 \)

Remark 12.23.

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 12.21 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 12.24.

Use your knowledge of derivatives of basic functions to complete Table 141 of antiderivatives. For each entry, your task is to find a function \(F\) whose derivative is the given function \(f\text{.}\)
Table 141. Familiar basic functions and their antiderivatives.
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 12.25.

Using this information, which of the following is an antiderivative for \(f(x) = 5\sin(x) - 4x^2\text{?}\)
  1. \(F(x) = -5\cos(x) +\frac{4}{3}x^3\text{.}\)
  2. \(F(x) = 5\cos(x) + \frac{4}{3}x^3\text{.}\)
  3. \(F(x) = -5\cos(x) - \frac{4}{3}x^3\text{.}\)
  4. \(F(x) = 5\cos(x) - \frac{4}{3}x^3\text{.}\)
Answer.
C. \(F(x)= -5 \cos (x)- \frac{4}{3} x^3 \text{.}\)

Activity 12.26.

Find the general antiderivative for each function.
(a)
\begin{equation*} f(x) = -4 \, \sec^2\left(x\right) \end{equation*}
Answer.
\(-4 \tan (x)+C \text{.}\)
(b)
\begin{equation*} f(x) = \frac{8}{\sqrt{x}} \end{equation*}
Answer.
\(16 \sqrt x +C \text{.}\)

Activity 12.27.

Find each indefinite integral.
(a)
\begin{equation*} \int (-9 \, x^{4} - 7 \, x^{2} + 4) \, dx \end{equation*}
Answer.
\(-\frac{9}{5} x^5 - \frac{7}{3} x^3+ 4x + C \)
(b)
\begin{equation*} \int 3 \, e^{x}\, dx \end{equation*}
Answer.
\(3 e^x + C \)

Subsection 12.3.2 Videos

Figure 142. Video for IN3

Subsection 12.3.3 Exercises

PrevTopNext