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Chapter 3: Applications of Differentiation
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Section 3.10: Antiderivatives
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Essentials



Introduction


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If $\frac{d}{\mathrm{dx}}F\left(x\right)\=f\left(x\right)$, then an antiderivative of $f\left(x\right)$ is $F\left(x\right)$. In other words, $F\left(x\right)$ is the function whose derivative is $f\left(x\right)$. Thus, given the derivative $f\left(x\right)$, the search for an antiderivative is the search for the function $F\left(x\right)$ for which the derivative is $f\left(x\right)$.

•

Since the derivatives of $F\left(x\right)$ and $F\left(x\right)\+c$, where $c$ is any constant, are the same function $f\left(x\right)$, the search for an antiderivative of $f\left(x\right)$ does not have a unique answer. There are many different antiderivatives of $f\left(x\right)$, but each differs from the other by an additive constant.

•

If any one antiderivative is sought, the search might be posed as "Find an antiderivative of $f\left(x\right)$," in which case any single antiderivative would suffice. Alternatively, if the complete family of all possible antiderivatives is sought, the search should be posed as "Find the (most general) antiderivative of $f\left(x\right)\,\"$ in which case any one antiderivative plus an arbitrary additive constant constitutes the complete family of all possible antiderivatives.

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There are calculus instructors who will interpret every request for an antiderivative as a request for the complete family, and will therefore insist that antiderivatives must always appear as $F\left(x\right)\+c$. On the other hand, there are calculus instructors who recognize the difference between a single member of the complete family and the complete family itself.

•

Table 3.10.1 lists an antiderivative for some common functions met while studying differentiation in Chapter 2. The table was constructed by gleaning from lists of derivatives, the functions that were differentiated. Note that neither this table, nor any of the common commercial tables, includes the additive constant. That is left for the user to include (or exclude).${}$

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Function

Antiderivative

Function

Antiderivative

$a$

$ax$

${\mathrm{sec}}^{}\left(x\right)\mathrm{tan}\left(x\right)$

$\mathrm{sec}\left(x\right)$

${x}^{1}$

$\mathrm{ln}\left(\leftx\right\right)$

$\mathrm{csc}\left(x\right)\mathrm{cot}\left(x\right)$

$\mathrm{csc}\left(x\right)$

${x}^{a}\,x\ne 1$

$\frac{{x}^{a\+1}}{a\+1}$

$\frac{1}{\sqrt{1{x}^{2}}}\,\leftx\right<1$

$\{\begin{array}{c}\mathrm{arcsin}\left(x\right)\\ \mathrm{arccos}\left(x\right)\end{array}$

$\mathrm{sin}\left(x\right)$

$\mathrm{cos}\left(x\right)$

$\frac{1}{1\+{x}^{2}}$

$\{\begin{array}{c}\mathrm{arctan}\left(x\right)\\ \mathrm{arccot}\left(x\right)\end{array}$

$\mathrm{cos}\left(x\right)$

$\mathrm{sin}\left(x\right)$

$\frac{1}{\sqrt{1\+{x}^{2}}}$

$\mathrm{arcsinh}\left(x\right)$

$\mathrm{tan}\left(x\right)$

$\mathrm{ln}\left(\left\mathrm{cos}\left(x\right)\right\right)$

$\frac{1}{1{x}^{2}}$

$\{\begin{array}{c}\mathrm{arctanh}\left(x\right)\,\leftx\right<1\\ \mathrm{arccoth}\left(x\right)\,\leftx\right\>1\end{array}$

$\mathrm{cot}\left(x\right)$

$\mathrm{ln}\left(\left\mathrm{sin}\left(x\right)\right\right)$

${e}^{x}$

${e}^{x}$

${\mathrm{sec}}^{2}\left(x\right)$

$\mathrm{tan}\left(x\right)$

$\mathrm{sinh}\left(x\right)$

$\mathrm{cosh}\left(x\right)$

${\mathrm{csc}}^{2}\left(x\right)$

$\mathrm{cot}\left(x\right)$

$\mathrm{cosh}\left(x\right)$

$\mathrm{sinh}\left(x\right)$

Table 3.10.1 Common antiderivatives



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Comments on Table 3.10.1


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The antiderivative of a constant $a$ is $ax$, even if $a\=0$, in which case the antiderivative is rightly given by $0\+c\=c$. In other words, the antiderivative of zero is any constant.

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The antiderivative of ${x}^{a}$ follows the "Power rule" provided $a\ne 1$. In words, the Power rule is "Add one to the power and divide by the new power." If $a\=1$, then the antiderivative is the logarithm. The typical calculus text will use absolute values, as in Table 3.10.1, but Maple's antiderivative for ${x}^{1}$ is just $\mathrm{ln}\left(x\right)$, a result that is correct on the complex plane.

•

An antiderivative of $1\/\sqrt{1{x}^{2}}$ can be given as either $\mathrm{arcsin}\left(x\right)$ or $\mathrm{arccos}\left(x\right)$. However, it should not be inferred from this that $\mathrm{arcsin}\left(x\right)\=\mathrm{arccos}\left(x\right)$. In fact, the correct identity is $\mathrm{arcsin}\left(x\right)\=\mathrm{\π}\/2\mathrm{arccos}\left(x\right)$. There are two forms for the antiderivative because the right additive constant transforms one antiderivative into the other.

•

The typical antiderivative tabulated for $1\/\left(1\+{x}^{2}\right)$ is $\mathrm{arctan}\left(x\right)$. The identity $\mathrm{arctan}\left(x\right)\=\mathrm{\π}\/2\mathrm{arccot}\left(x\right)$ and the additive constant argument accounts for the two different forms of the antiderivative.

•

It is surprising that the derivatives of $\mathrm{arctanh}\left(x\right)$ and $\mathrm{arccoth}\left(x\right)$ are the same. However, Figure 3.10.1 shows that the domains for these two antiderivatives are different.


Figure 3.10.1 Graphs of $\mathrm{arctanh}\left(x\right)$ (black) and $\mathrm{arccoth}\left(x\right)$ (red)



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Finally, it is worth remembering that an antiderivative for ${e}^{x}$ is ${e}^{x}$, that is, the exponential function is "its own derivative and its own antiderivative."



Antiderivatives in Maple


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Table 3.10.2 provides a simple tool for obtaining $F\left(x\right)$, the general antiderivative of $f\left(x\right)$. The arbitrary constant $\mathrm{\_C}$ is added to a basic antiderivative to give the complete family of antiderivatives. The underscore in front of the "C" indicates that Maple has generated that symbol. To use the tool, simply enter an expression in the box to the right of "$f\left(x\right)\=$" and press the button labeled "F(x) =".
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$f\left(x\right)\=$

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Table 3.10.2 A tool for obtaining antiderivatives



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Maple's Student Calculus1 package has two builtin tools that will, after a fashion, return an antiderivative. These are listed in Table 3.10.3.
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Table 3.10.4 contains illustrations of the AntiderivativePlot command used to return just an antiderivative.
$\mathrm{Student}:\mathrm{Calculus1}:\mathrm{AntiderivativePlot}\left(x\,\mathrm{output}\=\mathrm{antiderivative}\right)$ = $\frac{{1}}{{2}}{}{{x}}^{{2}}{}{50}$${}$

$\mathrm{Student}:\mathrm{Calculus1}:\mathrm{AntiderivativePlot}\left(x\,x\=0..3\,\mathrm{output}\=\mathrm{antiderivative}\right)$ = $\frac{{1}}{{2}}{}{{x}}^{{2}}$${}$

Table 3.10.4 The AntiderivativePlot command used to return an antiderivative



In the first instance, the default range $\left[a\,b\right]\=\left[10\,10\right]$ is active; hence, the additive constant $c$ in $F\left(x\right)\={x}^{2}\/2\+c$ is chosen (by default) so that $F\left(10\right)\=0$. In the second instance, the active interval is $\left[0\,3\right]$; the additive constant $c$ is chosen so that $F\left(0\right)\=0$. Of course, if the Student Calculus1 package is first loaded (Tools≻Load Package), the prefix "Student:Calculus1:" can be deleted from the AntiderivativePlot command.
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The AntiderivativePlot command can also return a graph showing the function $f\left(x\right)$, an antiderivative $F\left(x\right)$, and perhaps members of the complete family $F\left(x\right)\+c$. Again, the default interval $\left[a\,b\right]$ is $\left[10\,10\right]$. The commands that generate for Figures 3.10.2 and 3.10.3 are given in Table 3.10.5.${}$
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Figure 3.10.2 Graph of $f\left(x\right)\=x$ and $F\left(x\right)\={x}^{2}\/22$





Figure 3.10.3 Graph of $f\left(x\right)\=x$, $F\left(x\right)\={x}^{2}\/22$, and members of the family ${x}^{2}\/2\+c$






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In Figures 3.10.2 and 3.10.3, the interval $\left[a\,b\right]$ is $\left[2\,2\right]$. Hence, the antiderivative displayed is $F\left(x\right)\={x}^{2}\/22$, for which $F\left(2\right)\=0$. In Figure 3.10.3, the green curves are members of the family ${x}^{2}\/2\+c$.
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Figure

Syntax for AntiderivativePlot Command

3.10.2

$\mathrm{Student}:\mathrm{Calculus1}:\mathrm{AntiderivativePlot}\left(x\,x\=2..2\right)$

3.10.3

$\mathrm{Student}:\mathrm{Calculus1}:\mathrm{AntiderivativePlot}\left(x\,x\=2..2\,\mathrm{showclass}\right)$

Table 3.10.5 Syntax for generating Figures 3.10.2 and 3.10.3



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Figure 3.10.4 shows the
tutor applied to $f\left(x\right)\=x$ on a default interval $\left[2\,2\right]$. The displayed antiderivative $F\left(x\right)\={x}^{2}\/22$ is obtained by choosing the additive constant $c$ so that $F\left(2\right)\=0$. To see members of the complete family of antiderivatives, check the "Show class of antiderivatives" box and then click the Display button.
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Figure 3.10.4 Screen image of the Antiderivative tutor applied to $f\left(x\right)\=x$ on the interval $\left[2\,2\right]$



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Examples


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Example 3.10.1

If $f\left(x\right)\=3{x}^{2}6\sqrt{x}plus;5$, find the antiderivative $F\left(x\right)$ for which $F\left(4\right)\=15$.

Example 3.10.2

A particle moving along the $x$axis has acceleration $x\u2033\left(t\right)\=5tplus;3$, initial velocity $x\prime \left(0\right)\=7$, and initial position $x\left(0\right)\=8$. Find $x\left(t\right)$, its position function.

Example 3.10.3

If $g\u2033\left(x\right)\=4{x}^{2}9x5$, find the antiderivative $g\left(x\right)$ for which $g\left(1\right)\=3$ and $g\left(3\right)\=1$.

Example 3.10.4

From the edge of a cliff 500 ft above ground level, a rock is thrown upward with a speed of 60 ft/s.
a)

Obtain the position function $y\left(t\right)$.

b)

When does the rock reach its maximum height?

c)

What is the maximum height attained?

d)

When does the rock drop to ground level?


Example 3.10.5

A car makes a panic stop with a constant deceleration of $30\mathrm{ft}sol;{\mathrm{s}}^{2}$ and leaves a skid mark of 180 ft. How fast was the car traveling when the brakes were first applied?



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