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<quiz>
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    <name>
      <text>CR-Int-14-substitution-6.b</text>
    </name>
    <questiontext format="html">
      <text>Using the method of integration by substitution, find the following integral:\[ \int {@p@} d{@v@}. \]

[[input:ans1]] [[validation:ans1]]</text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[We need to calculate \[\int {@p@} d{@v@}\] by means of the substitution \(u={@pp1@}\) [[facts:calc_int_methods_substitution]].

So, if \(u = {@pp1@}\) then \(d{@u@} = {@diff(pp1,v)@} d{@v@}\), so, \(d{@v@} = d{@u@}/{@diff(pp1,v)@}\).

Hence, 

\[\begin{array}{rcl}\displaystyle  
\int {@p@} d{@v@} &\displaystyle =&\displaystyle  \int {@pp2/u@} \frac{d{@u@}}{@diff(pp1,v)@} \\&\displaystyle =&\displaystyle  \int {@pp2/u/diff(pp1,v)@} d{@u@} \\&\displaystyle =&\displaystyle  {@pp2/diff(pp1,v)@} \int {@1/u@} d{@u@} \\&\displaystyle =&\displaystyle  {@pp2/diff(pp1,v)@}{@int(1/u,u)+c@} \\&\displaystyle =&\displaystyle  {@int(p,v)+c@}
 \end{array}\]]]></text>
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    <stackversion>
      <text>2019121800</text>
    </stackversion>
    <questionvariables>
      <text>v:x;
rdm1:1+rand(5);
rdm2:2+rand(4);
pp1:rdm1+cos(rdm2*v);
pp2:sin(rdm2*v);
p:pp2/pp1;
dumvar6:u;
ta1:int(p,v)+c;</text>
    </questionvariables>
    <specificfeedback format="html">
      <text><![CDATA[<p>[[feedback:Result]]</p>]]></text>
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    <questionnote format="moodle_auto_format">
      <text>\[\int {@p@}d{@v@} = {@int(p,v)@}+c\]</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:red;"><i class="fa fa-times"></i></span> Incorrect answer.]]></text>
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