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    <name>
      <text>13.4.2.2 T Definite integral by parts</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find \(\displaystyle \int_0^{{@ub@}} {@exp@} \, \mathrm{d}x\) to \(3\) d.p.</p>
<p>What are your choices for \(f\) and \(g\)?</p>
<p>\(f(x) = \) [[input:ans1]] [[validation:ans1]] [[feedback:prt1]]</p>
<p>\(g(x) = \) [[input:ans2]] [[validation:ans2]] [[feedback:prt2]]</p>
<p>Now complete the integral.</p>
<p>\(\displaystyle \int_0^{{@ub@}} {@exp@} \, \mathrm{d}x = \) [[input:ans3]] [[validation:ans3]][[feedback:prt3]]</p>]]></text>
    </questiontext>
    <generalfeedback format="html">
      <text><![CDATA[<p>We are asked to find \(\displaystyle \int_0^{{@ub@}} {@exp@} \, \mathrm{d}x\). In order to do this we use integration by parts.</p>
<p>We must choose \(f(x) = {@f@}\) and \(g(x) = {@g@}\).</p>
<p>Now we use the integration by parts formula: \[\int_a^b f \cdot g \, \mathrm{d}x = \left[f \cdot \int g \, \mathrm{d}x \right]_a^b - \int_a^b \left(\dfrac{df}{dx} \int g \, \mathrm{d}x \right) \, \mathrm{d}x\]
</p>
<p>First we find \(\displaystyle \frac{df}{dx}\) and \(\displaystyle \int g(x) \, \mathrm{d}x\). \[\begin{aligned} \frac{df}{dx} &amp;= {@df@} \\ \int g(x)\,\mathrm{d}x &amp;= {@ig@} \end{aligned}\]
</p>
<p>The formula then gives \[\begin{aligned} \int_0^{{@ub@}} {@exp@} \,\mathrm{d}x &amp;= \Bigg[\left({@f@}\right) \cdot \left({@ig@}\right)\Bigg]_0^{{@ub@}} - \int_0^{{@ub@}} \left({@df@}\right) \cdot \left({@ig@}\right) \,\mathrm{d}x\\ &amp;= \Bigg[{@f * ig@}\Bigg]_0^{{@ub@}}
    - \Bigg[{@idfig@}\Bigg]_0^{{@ub@}} \\ &amp;= \left({@u2@} - \left({@u1@}\right)\right) - \left({@v2@} - \left({@v1@}\right)\right) \\ &amp;= {@ta3@} \end{aligned}\]
</p>]]></text>
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    <stackversion>
      <text>2020120600</text>
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    <questionvariables>
      <text>a: 2 + rand(3);
b: 2 + rand(4);
ub: %pi / 4;

f: a*x + b;
g: sin(x);
exp: f * g;

/* Needed for solution */
df: diff(f, x);
ig: int(g, x);
fig: f * ig;
idfig: int(df * ig, x);
u1: subst(0, x, fig);
u2: subst(ub, x, fig);
v1: subst(0, x, idfig);
v2: subst(ub, x, idfig);

/* Teacher's answers */
ta1: f;
ta2: g;
ta3: decimalplaces(float(int(exp, x, 0, ub)), 3);</text>
    </questionvariables>
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    <questionnote format="moodle_auto_format">
      <text>\(\int_0^{{@ub@}} {@exp@} \,\mathrm{d}x = {@ta3@}\)</text>
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      <text><![CDATA[<span style="font-size: 1.5em; color:green;"><i class="fa fa-check"></i></span> Correct answer, well done.]]></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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