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<html xmlns="http://www.w3.org/1999/xhtml"
xmlns:xlink="http://www.w3.org/1999/xlink" >
<head>
<title>Section8.8</title>
<link rel="stylesheet" type="text/css" href="graphicstyle.css" />
<script type="text/javascript" src="bookax1.js" />
</head>
<body>
<a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a><a href="section-8.7.xhtml" style="margin-right: 10px;">Previous Section 8.7 Laplace Transforms</a><a href="section-8.9.xhtml" style="margin-right: 10px;">Next Section 8.9 Working with Power Series</a>
<a href="book-index.xhtml">Book Index</a><div class="section" id="sec-8.8">
<h2 class="sectiontitle">8.8 Integration</h2>
<a name="ugProblemIntegration" class="label"/>
<p>Integration is the reverse process of differentiation, that is,
<span class="index">integration</span><a name="chapter-8-98"/> an <span class="italic">integral</span> of a function <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math> with respect
to a variable <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math> is any function <math xmlns="&mathml;" mathsize="big"><mstyle><mi>g</mi></mstyle></math> such that <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>D</mi><mo>(</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> is equal to
<math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math>.
</p>
<p>The package <span class="teletype">FunctionSpaceIntegration</span> provides the top-level
integration operation, <span class="spadfunFrom" >integrate</span><span class="index">integrate</span><a name="chapter-8-99"/><span class="index">FunctionSpaceIntegration</span><a name="chapter-8-100"/>,
for integrating real-valued elementary functions.
<span class="index">FunctionSpaceIntegration</span><a name="chapter-8-101"/>
</p>
<div id="spadComm8-117" class="spadComm" >
<form id="formComm8-117" action="javascript:makeRequest('8-117');" >
<input id="comm8-117" type="text" class="command" style="width: 22em;" value="integrate(cosh(a*x)*sinh(a*x), x)" />
</form>
<span id="commSav8-117" class="commSav" >integrate(cosh(a*x)*sinh(a*x), x)</span>
<div id="mathAns8-117" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mrow><msup><mrow><mo>sinh</mo><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>x</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mrow><mo>cosh</mo><mo>(</mo><mrow><mi>a</mi><mo></mo><mi>x</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Expression Integer,...)
</div>
<p>Unfortunately, antiderivatives of most functions cannot be expressed in
terms of elementary functions.
</p>
<div id="spadComm8-118" class="spadComm" >
<form id="formComm8-118" action="javascript:makeRequest('8-118');" >
<input id="comm8-118" type="text" class="command" style="width: 28em;" value="integrate(log(1 + sqrt(a * x + b)) / x, x)" />
</form>
<span id="commSav8-118" class="commSav" >integrate(log(1 + sqrt(a * x + b)) / x, x)</span>
<div id="mathAns8-118" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><msup><mo>∫</mo><mrow><mi>x</mi></mrow></msup><mrow><mfrac><mrow><mo>log</mo><mo>(</mo><mrow><mrow><msqrt><mrow><mi>b</mi><mo>+</mo><mrow><mo>%</mo><mi>M</mi><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>%</mo><mi>M</mi></mrow></mfrac><mo></mo><mrow><mi>d</mi><mo>%</mo><mi>M</mi></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Expression Integer,...)
</div>
<p>Given an elementary function to integrate, Axiom returns a formal
integral as above only when it can prove that the integral is not
elementary and not when it cannot determine the integral.
In this rare case it prints a message that it cannot
determine if an elementary integral exists.
</p>
<p>Similar functions may have antiderivatives <span class="index">antiderivative</span><a name="chapter-8-102"/>
that look quite different because the form of the antiderivative
depends on the sign of a constant that appears in the function.
</p>
<div id="spadComm8-119" class="spadComm" >
<form id="formComm8-119" action="javascript:makeRequest('8-119');" >
<input id="comm8-119" type="text" class="command" style="width: 17em;" value="integrate(1/(x**2 - 2),x)" />
</form>
<span id="commSav8-119" class="commSav" >integrate(1/(x**2 - 2),x)</span>
<div id="mathAns8-119" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo></mo><mrow><msqrt><mn>2</mn></msqrt></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>x</mi></mrow></mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>-</mo><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mrow><mn>2</mn><mo></mo><mrow><msqrt><mn>2</mn></msqrt></mrow></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Expression Integer,...)
</div>
<div id="spadComm8-120" class="spadComm" >
<form id="formComm8-120" action="javascript:makeRequest('8-120');" >
<input id="comm8-120" type="text" class="command" style="width: 17em;" value="integrate(1/(x**2 + 2),x)" />
</form>
<span id="commSav8-120" class="commSav" >integrate(1/(x**2 + 2),x)</span>
<div id="mathAns8-120" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>arctan</mo><mo>(</mo><mfrac><mrow><mi>x</mi><mo></mo><mrow><msqrt><mn>2</mn></msqrt></mrow></mrow><mn>2</mn></mfrac><mo>)</mo></mrow><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(Expression Integer,...)
</div>
<p>If the integrand contains parameters, then there may be several possible
antiderivatives, depending on the signs of expressions of the parameters.
</p>
<p>In this case Axiom returns a list of answers that cover all the
possible cases. Here you use the answer involving the square root of
<math xmlns="&mathml;" mathsize="big"><mstyle><mi>a</mi></mstyle></math> when <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo>></mo><mn>0</mn></mrow></mstyle></math> and <span class="index">integration:result as list of real
functions</span><a name="chapter-8-103"/> the answer involving the square root of <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mo>-</mo><mi>a</mi></mrow></mstyle></math> when <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo><</mo><mn>0</mn></mrow></mstyle></math>.
</p>
<div id="spadComm8-121" class="spadComm" >
<form id="formComm8-121" action="javascript:makeRequest('8-121');" >
<input id="comm8-121" type="text" class="command" style="width: 23em;" value="integrate(x**2 / (x**4 - a**2), x)" />
</form>
<span id="commSav8-121" class="commSav" >integrate(x**2 / (x**4 - a**2), x)</span>
<div id="mathAns8-121" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mtable><mtr><mtd><mrow><mo>[</mo><mfrac><mrow><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>a</mi><mo></mo><mi>x</mi></mrow></mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>-</mo><mi>a</mi></mrow></mfrac><mo>)</mo></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mfrac><mrow><mi>x</mi><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow><mi>a</mi></mfrac><mo>)</mo></mrow></mrow></mrow><mrow><mn>4</mn><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow></mfrac><mo>,</mo></mrow></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mo></mo><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>a</mi><mo></mo><mi>x</mi></mrow></mrow><mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>a</mi></mrow></mfrac><mo>)</mo></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mfrac><mrow><mi>x</mi><mo></mo><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow></mrow><mi>a</mi></mfrac><mo>)</mo></mrow></mrow></mrow><mrow><mn>4</mn><mo></mo><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow></mrow></mfrac><mo>]</mo></mrow></mtd></mtr></mtable></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(List Expression Integer,...)
</div>
<p>If the parameters and the variables of integration can be complex
numbers rather than real, then the notion of sign is not defined. In
this case all the possible answers can be expressed as one complex
function. To get that function, rather than a list of real functions,
use <span class="spadfunFrom" >complexIntegrate</span><span class="index">complexIntegrate</span><a name="chapter-8-104"/><span class="index">FunctionSpaceComplexIntegration</span><a name="chapter-8-105"/>,
which is provided by the package <span class="index">integration:result as a
complex functions</span><a name="chapter-8-106"/> <span class="teletype">FunctionSpaceComplexIntegration</span>.
<span class="index">FunctionSpaceComplexIntegration</span><a name="chapter-8-107"/>
</p>
<p>This operation is used for integrating complex-valued elementary
functions.
</p>
<div id="spadComm8-122" class="spadComm" >
<form id="formComm8-122" action="javascript:makeRequest('8-122');" >
<input id="comm8-122" type="text" class="command" style="width: 28em;" value="complexIntegrate(x**2 / (x**4 - a**2), x)" />
</form>
<span id="commSav8-122" class="commSav" >complexIntegrate(x**2 / (x**4 - a**2), x)</span>
<div id="mathAns8-122" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow><mo></mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mi>x</mi><mo></mo><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mrow><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow><mo></mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mi>x</mi><mo></mo><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mrow><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow><mo></mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mi>x</mi><mo></mo><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mrow><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow><mo></mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mi>x</mi><mo></mo><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mrow><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn><mo></mo><mrow><msqrt><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow><mo></mo><mrow><msqrt><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow></msqrt></mrow></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>As with the real case, antiderivatives for most complex-valued
functions cannot be expressed in terms of elementary functions.
</p>
<div id="spadComm8-123" class="spadComm" >
<form id="formComm8-123" action="javascript:makeRequest('8-123');" >
<input id="comm8-123" type="text" class="command" style="width: 33em;" value="complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)" />
</form>
<span id="commSav8-123" class="commSav" >complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)</span>
<div id="mathAns8-123" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><msup><mo>∫</mo><mrow><mi>x</mi></mrow></msup><mrow><mfrac><mrow><mo>log</mo><mo>(</mo><mrow><mrow><msqrt><mrow><mi>b</mi><mo>+</mo><mrow><mo>%</mo><mi>M</mi><mo></mo><mi>a</mi></mrow></mrow></msqrt></mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>%</mo><mi>M</mi></mrow></mfrac><mo></mo><mrow><mi>d</mi><mo>%</mo><mi>M</mi></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>Sometimes <span style="font-weight: bold;"> integrate</span> can involve symbolic algebraic numbers
such as those returned by <span class="spadfunFrom" >rootOf</span><span class="index">rootOf</span><a name="chapter-8-108"/><span class="index">Expression</span><a name="chapter-8-109"/>.
To see how to work with these strange generated symbols (such as
<math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mo>%</mo><mo>%</mo><mi>a0</mi></mrow></mstyle></math>), see
<a href="section-8.3.xhtml#ugxProblemSymRootAll" class="ref" >ugxProblemSymRootAll</a> .
</p>
<p>Definite integration is the process of computing the area between
<span class="index">integration:definite</span><a name="chapter-8-110"/>
the <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>-axis and the curve of a function <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math>.
The fundamental theorem of calculus states that if <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math> is
continuous on an interval <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo>.</mo><mo>.</mo><mi>b</mi></mrow></mstyle></math> and if there exists a function <math xmlns="&mathml;" mathsize="big"><mstyle><mi>g</mi></mstyle></math>
that is differentiable on <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo>.</mo><mo>.</mo><mi>b</mi></mrow></mstyle></math> and such that <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>D</mi><mo>(</mo><mi>g</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math>
is equal to <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math>, then the definite integral of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math>
for <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math> in the interval <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo>.</mo><mo>.</mo><mi>b</mi></mrow></mstyle></math> is equal to <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>g</mi><mo>(</mo><mi>b</mi><mo>)</mo><mo>-</mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mstyle></math>.
</p>
<p>The package <span class="teletype">RationalFunctionDefiniteIntegration</span> provides
the top-level definite integration operation,
<span class="spadfunFrom" >integrate</span><span class="index">integrate</span><a name="chapter-8-111"/><span class="index">RationalFunctionDefiniteIntegration</span><a name="chapter-8-112"/>,
for integrating real-valued rational functions.
</p>
<div id="spadComm8-124" class="spadComm" >
<form id="formComm8-124" action="javascript:makeRequest('8-124');" >
<input id="comm8-124" type="text" class="command" style="width: 42em;" value="integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)" />
</form>
<span id="commSav8-124" class="commSav" >integrate((x**4 - 3*x**2 + 6)/(x**6-5*x**4+5*x**2+4), x = 1..2)</span>
<div id="mathAns8-124" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mrow><mo>arctan</mo><mo>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>)</mo></mrow></mrow><mo>-</mo><mi>π</mi></mrow><mn>2</mn></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(f1: OrderedCompletion Expression Integer,...)
</div>
<p>Axiom checks beforehand that the function you are integrating is
defined on the interval <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>a</mi><mo>.</mo><mo>.</mo><mi>b</mi></mrow></mstyle></math>, and prints an error message if it
finds that this is not case, as in the following example:
</p>
<div class="verbatim"><br />
integrate(1/(x**2-2), x = 1..2)<br />
<br />
>> Error detected within library code:<br />
Pole in path of integration<br />
You are being returned to the top level<br />
of the interpreter.<br />
</div>
<p>When parameters are present in the function, the function may or may not be
defined on the interval of integration.
</p>
<p>If this is the case, Axiom issues a warning that a pole might
lie in the path of integration, and does not compute the integral.
</p>
<div id="spadComm8-125" class="spadComm" >
<form id="formComm8-125" action="javascript:makeRequest('8-125');" >
<input id="comm8-125" type="text" class="command" style="width: 21em;" value="integrate(1/(x**2-a), x = 1..2)" />
</form>
<span id="commSav8-125" class="commSav" >integrate(1/(x**2-a), x = 1..2)</span>
<div id="mathAns8-125" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mi>potentialPole</mi></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(pole: potentialPole,...)
</div>
<p>If you know that you are using values of the parameter for which
the function has no pole in the interval of integration, use the
string <span class="teletype">``noPole''</span> as a third argument to
<span class="spadfunFrom" >integrate</span><span class="index">integrate</span><a name="chapter-8-113"/><span class="index">RationalFunctionDefiniteIntegration</span><a name="chapter-8-114"/>:
</p>
<p>The value here is, of course, incorrect if <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>sqrt</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mstyle></math> is between
<math xmlns="&mathml;" mathsize="big"><mstyle><mn>1</mn></mstyle></math> and <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mn>2</mn><mo>.</mo></mrow></mstyle></math>
</p>
<div id="spadComm8-126" class="spadComm" >
<form id="formComm8-126" action="javascript:makeRequest('8-126');" >
<input id="comm8-126" type="text" class="command" style="width: 28em;" value='integrate(1/(x**2-a), x = 1..2, "noPole")' />
</form>
<span id="commSav8-126" class="commSav" >integrate(1/(x**2-a), x = 1..2, "noPole")</span>
<div id="mathAns8-126" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>(</mo><mo>-</mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mo>-</mo><mrow><mn>4</mn><mo></mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow><mo>+</mo><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mi>a</mi></mrow><mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow><mo>+</mo><mrow><mo>log</mo><mo>(</mo><mfrac><mrow><mrow><mrow><mo>(</mo><mo>-</mo><mrow><mn>8</mn><mo></mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>32</mn><mo></mo><mi>a</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow><mo>+</mo><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>+</mo><mrow><mn>24</mn><mo></mo><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>16</mn><mo></mo><mi>a</mi></mrow></mrow><mrow><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mo>-</mo><mrow><mn>8</mn><mo></mo><mi>a</mi></mrow><mo>+</mo><mn>16</mn></mrow></mfrac><mo>)</mo></mrow><mfrac><mrow><mo>-</mo><mrow><mo>arctan</mo><mo>(</mo><mfrac><mrow><mn>2</mn><mo></mo><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow></mrow><mi>a</mi></mfrac><mo>)</mo></mrow><mo>+</mo><mrow><mo>arctan</mo><mo>(</mo><mfrac><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow><mi>a</mi></mfrac><mo>)</mo></mrow></mrow><mrow><msqrt><mrow><mo>-</mo><mi>a</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mrow><mn>4</mn><mo></mo><mrow><msqrt><mi>a</mi></msqrt></mrow></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Union(f2: List OrderedCompletion Expression Integer,...)
</div>
</div><a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a>
<a href="section-8.7.xhtml" style="margin-right: 10px;">Previous Section 8.7 Laplace Transforms</a><a href="section-8.9.xhtml" style="margin-right: 10px;">Next Section 8.9 Working with Power Series</a>
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