<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>CR-Int-15-partialfrac-4.b</text>
    </name>
    <questiontext format="html">
      <text>By expressing the integrand as the sum of its partial fractions, find the following definite integral:
\[ \int^{@rdm4@}_{@rdm5@} {@p@} d{@v@}. \]

[[input:ans1]] [[validation:ans1]]</text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[In order to calculate \[\int {@p@} d{@v@}\] we need to express \[{@p@}\] as the sum of its partial fractions <hint>alg_partial_fractions</hint>.

\[\begin{array}{rcl}\displaystyle 
 {@p@} &\displaystyle =&\displaystyle  {@factor(p)@} \\
   &\displaystyle =&\displaystyle  {@A@}{@ppp1@} + {@B@}{@ppp2@}
 \end{array}\] 

Omitting details we have \[{@p@} = {@q@},\]
hence,

\[\begin{array}{rcl}\displaystyle 
 \int {@p@} d{@v@} &\displaystyle =&\displaystyle  \int {@q@} d{@v@} \\
   &\displaystyle =&\displaystyle  {@rdm2@} \int {@ppp1@} d{@v@} + {@rdm3@} \int {@ppp2@} d{@v@} \\
   &\displaystyle =&\displaystyle  {@rdm2*int(ppp1,v)@} + {@rdm3*int(ppp2,v)@} +c \\
   &\displaystyle =&\displaystyle  {@rdm2*int(ppp1,v)+rdm3*int(ppp2,v)@} +c
 \end{array}\] 

We need \[\int^{@rdm4@}_{@rdm5@} {@p@} d{@v@}\] so, recall that if \[\int f(x) dx = F(x)\] then \[\int^a_b f(x) dx = F(a) - F(b).\]

Thus, define \[F({@v@}) = {@int(q,v)@}\]

Then, 

\[\begin{array}{rcl}\displaystyle 
  \int^{@rdm4@}_{@rdm5@} {@p@} d{@v@} &\displaystyle =&\displaystyle  F({@rdm4@}) - F({@rdm5@}) \\
   &\displaystyle =&\displaystyle  {@subst(rdm4,v,pq)@} - \left[{@subst(rdm5,v,pq)@}\right] \\
   &\displaystyle =&\displaystyle  {@subst(rdm4,v,pq)-subst(rdm5,v,pq)@}
 \end{array}\]]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.1</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>v:x;
rdm1:9;
rdm2:2;
rdm3:37;
rdm4:1;
rdm5:-1;
ppp1:1/(v+rdm1);
ppp2:ppp1^2;
pp1:rdm2*ppp1;
pp2:rdm3*ppp2;
q:pp1+pp2;
p:simplify(factor(q));
dumvar7:A;
dumvar8:B;
pq:int(q,v);</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\[\int^{@rdm4@}_{@rdm5@} {@p@} d{@v@} = {@int(p,v,rdm5,rdm4)@}\]</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
    <questionsimplify>1</questionsimplify>
    <assumepositive>0</assumepositive>
    <assumereal>0</assumereal>
    <prtcorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></text>
    </prtcorrect>
    <prtpartiallycorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:orange;"><i class="fa fa-adjust"></i></span> Your answer is partially correct.]]></text>
    </prtpartiallycorrect>
    <prtincorrect format="html">
      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
    </prtincorrect>
    <decimals>.</decimals>
    <scientificnotation>*10</scientificnotation>
    <multiplicationsign>dot</multiplicationsign>
    <sqrtsign>1</sqrtsign>
    <complexno>i</complexno>
    <inversetrig>cos-1</inversetrig>
    <logicsymbol>lang</logicsymbol>
    <matrixparens>[</matrixparens>
    <isbroken>0</isbroken>
    <variantsselectionseed/>
    <input>
      <name>ans1</name>
      <type>algebraic</type>
      <tans>int(p,v,rdm5,rdm4)</tans>
      <boxsize>15</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
      <syntaxhint/>
      <syntaxattribute>0</syntaxattribute>
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      <requirelowestterms>1</requirelowestterms>
      <checkanswertype>1</checkanswertype>
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      <options/>
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      <name>Result</name>
      <value>1.0000000</value>
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        <text/>
      </feedbackvariables>
      <node>
        <name>0</name>
        <description/>
        <answertest>AlgEquiv</answertest>
        <sans>ans1</sans>
        <tans>int(p,v,rdm5,rdm4)</tans>
        <testoptions/>
        <quiet>0</quiet>
        <truescoremode>=</truescoremode>
        <truescore>1.0000000</truescore>
        <truepenalty/>
        <truenextnode>-1</truenextnode>
        <trueanswernote>Result-0-T </trueanswernote>
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          <text/>
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          <text/>
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    </prt>
    <qtest>
      <testcase>1</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>int(p,v,rdm5,rdm4)</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>1.0000000</expectedscore>
        <expectedpenalty>0.0000000</expectedpenalty>
        <expectedanswernote>Result-0-T</expectedanswernote>
      </expected>
    </qtest>
    <qtest>
      <testcase>2</testcase>
      <description/>
      <testinput>
        <name>ans1</name>
        <value>ev(int(p,v)+c,simp)</value>
      </testinput>
      <expected>
        <name>Result</name>
        <expectedscore>0.0000000</expectedscore>
        <expectedpenalty>0.1000000</expectedpenalty>
        <expectedanswernote>Result-0-F</expectedanswernote>
      </expected>
    </qtest>
  </question>
</quiz>
