// Numbas version: exam_results_page_options
{"name": "Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]</p>\n\t\t\t      <p>$I=\\;$[[0]]</p>\n\t\t\t      <p>Input numbers in your answer as integers or fractions and not as decimals.</p>\n\t\t\t      <p>Click on Show steps to get further help. You will lose 1 mark if you do so.</p>\n\t\t\t      \n\t\t\t", "marks": 0.0, "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$</p>", "type": "information", "marks": 0.0}], "stepspenalty": 1.0, "gaps": [{"checkingtype": "absdiff", "answersimplification": "std", "checkingaccuracy": 0.001, "notallowed": {"strings": ["."], "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "partialcredit": 0.0, "showstrings": false}, "type": "jme", "vsetrange": [0.0, 1.0], "marks": 3.0, "vsetrangepoints": 5.0, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C"}], "type": "gapfill"}], "progress": "ready", "question_groups": [{"pickQuestions": 0, "name": "", "questions": [], "pickingStrategy": "all-ordered"}], "metadata": {"description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>", "notes": "\n\t\t    \t\t<p><strong>2/08/2012:</strong></p>\n\t\t    \t\t    <p>Added tags.</p>\n\t\t    \t\t    <p>Added description.</p>\n\t\t    \t\t    <p>Checked calculation. OK.</p>\n\t\t    \t\t    <p>Added information about Show steps in prompt content area.&nbsp;</p>\n\t\t    \t\t    <p>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n\t\t    \t\t    <p>Got rid of a redundant ruleset.</p>\n\t\t    \t\t    <p>&nbsp;</p>\n\t\t    \t\t    <p>&nbsp;</p>\n\t\t    \n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "name": "Julie's copy of Indefinite integral by substitution", "variables": {"a": {"name": "a", "definition": "random(1..5)"}, "m": {"name": "m", "definition": "random(4..9)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}, "b": {"name": "b", "definition": "s1*random(1..9)"}}, "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "statement": "\n\t  <p>Find the following integral.</p>\n\t  <p>Input the constant of integration as $C$.</p>\n\t  \n\t", "extensions": [], "advice": "\n\t    \n\t    \n\t    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$<br/>Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.</p>\n\t    \n\t    \n\t    \n\t    <p>Hence the integral becomes:</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\begin{eqnarray*} I&#38;=&#38;\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n\t    \n\t    \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t    \n\t    <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; f&#39;(x)g(f(x))\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f&#39;(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int g(u)\\;du \\]<br/>And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.</p>\n\t    \n\t    \n\t  \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "variable_groups": [], "functions": {}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}