<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="stack">
    <name>
      <text>Int_Reciprocal_Sqrt_2</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p>Complete the square in the quadratic expression q which is given by \[ q= {@q1@}, \] and use this to evaluate the following integral \[ \int {@f1@}d{@v1@} \] [[input:ans1]] [[validation:ans1]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>Completing the square for \(q\) gives 
\[q={@n1^2@}{@v1*v1@} + {@2*n0*n1@}{@v1@} + {@n0^2+c0^2@}={@n1^2@}{@v1*v1@}+ {@2*n1*n0@}{@v1@}+{@n0^2@}+{@c0^2@}=({@n1@}{@v1@}+{@n0@})^2 + {@c0^2@}\]</p>
<p>Therefore \[\int{@f1@}d{@v1@} = \int\frac{1}{@sqrt((n1*v1+n0)^2 + c0^2)@}d{@v1@}=\int\frac{1}{@sqrt(c0^2*((n1*v1/c0+n0/c0)^2 +1))@}d{@v1@}\]</p>
<p> </p>
<p>Let \({@n1*v1/c0@}+{@n0/c0@} = \sinh(u)\)</p>
<p>giving \[d{@v1@} = {@c0/n1*cosh(u)@}du\]</p>
<p>Therefore \[\int{@f1@}d{@v1@} = \int\frac{1}{@c0*sqrt(sinh(u)^2+1)@}{@c0/n1*cosh(u)@}du\]</p>
<p>and since \(\cosh^2(u)-\sinh^2(u) = 1\)</p>
<p>\[\begin{array}{rcl}\displaystyle \int\frac{1}{@sqrt(sinh(u)^2+1)@}{@cosh(u)/n1@}du & \displaystyle =&\displaystyle\int\frac{1}{@cosh(u)@}{@cosh(u)/n1@}du\\  \int \frac{du}{@n1@} & \displaystyle =& \displaystyle \frac{u}{@n1@} + c \\ & \displaystyle =&\displaystyle {@correctanswer@} \end{array}\]</p>]]></text>
    </generalfeedback>
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    <idnumber/>
    <stackversion>
      <text/>
    </stackversion>
    <questionvariables>
      <text>n0:rand(5)+1;
 n1:rand(4)+2;
 c0:rand(3)+2;
 v1:x;
 q1:expand((n1*v1+n0)^2 + c0^2);
 f1:1/sqrt(q1);
 correctanswer:expand(int(f1,v1))+c;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
    </specificfeedback>
    <questionnote format="moodle_auto_format">
      <text>\( \int {@f1@} d{@v1@}\) gives {@correctanswer@}</text>
    </questionnote>
    <questiondescription format="moodle_auto_format">
      <text/>
    </questiondescription>
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    <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>
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      <type>algebraic</type>
      <tans>correctanswer</tans>
      <boxsize>20</boxsize>
      <strictsyntax>1</strictsyntax>
      <insertstars>0</insertstars>
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        <answertest>AlgEquiv</answertest>
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      <description/>
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  </question>
</quiz>
