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Calculus for Team-Based Inquiry Learning 2025 Edition

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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 \(g\text{.}\)
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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*}
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Activity 4.3.2.

Consider the function \(f(x)=\cos x\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
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  1. \(\displaystyle \sin x\)
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  2. \(\displaystyle \cos x\)
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  3. \(\displaystyle \tan x\)
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  4. \(\displaystyle \sec x\)
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Activity 4.3.3.

Consider the function \(f(x)=x^2\text{.}\) Which of the following could be \(F(x)\text{,}\) an antiderivative of \(f(x)\text{?}\)
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  1. \(\displaystyle 2x \)
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  2. \(\displaystyle \frac{1}{3}x^3 \)
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  3. \(\displaystyle x^3\)
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  4. \(\displaystyle \frac{2}{3}x^3 \)
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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.
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In the following activity, we work to build a list of basic functions whose antiderivatives we already know.
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Activity 4.3.5.

Use your knowledge of derivatives of basic functions to complete Table 85 of antiderivatives. For each entry, your task is to find a function \(F\) whose derivative is the given function \(f\text{.}\)
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Table 85. Familiar basic functions and their antiderivatives.
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given function, \(f(x)\) antiderivative, \(F(x)\)  
\(k\text{,}\) (\(k\) is constant)
\(x^n\text{,}\) \(n \ne -1\)
\(\frac{1}{x}\text{,}\) \(x \gt 0\)
\(\sin(x)\)
\(\cos(x)\)
\(\sec(x) \tan(x)\)
\(\csc(x) \cot(x)\)
\(\sec^2 (x)\)
\(\csc^2 (x)\)
\(e^x\)
\(a^x\) \((a \gt 1)\)
\(\frac{1}{1+x^2}\)
\(\frac{1}{\sqrt{1-x^2}}\)
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Activity 4.3.6.

Using this information, which of the following is an antiderivative for \(f(x) = 5\sin(x) - 4x^2\text{?}\)
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  1. \(F(x) = -5\cos(x) +\frac{4}{3}x^3\text{.}\)
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  2. \(F(x) = 5\cos(x) + \frac{4}{3}x^3\text{.}\)
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  3. \(F(x) = -5\cos(x) - \frac{4}{3}x^3\text{.}\)
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  4. \(F(x) = 5\cos(x) - \frac{4}{3}x^3\text{.}\)
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Activity 4.3.7.

Find the general antiderivative for each function.
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(a)
\begin{equation*} f(x) = -4 \, \sec^2\left(x\right) \end{equation*}
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(b)
\begin{equation*} f(x) = \frac{8}{\sqrt{x}} \end{equation*}
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Activity 4.3.8.

Find each indefinite integral.
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(a)
\begin{equation*} \int (-9 \, x^{4} - 7 \, x^{2} + 4) \, dx \end{equation*}
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(b)
\begin{equation*} \int 3 \, e^{x}\, dx \end{equation*}
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Subsection 4.3.2 Videos

Figure 86. Video for IN3
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Subsection 4.3.3 Exercises

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