// Numbas version: finer_feedback_settings
{"name": "Blathnaid's copy of cormac's copy of 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": [{"name": "Blathnaid's copy of cormac's copy of Julie's copy of Indefinite integral by substitution", "tags": [], "metadata": {"description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the following integral using <strong>integration by substitution</strong>.</p>\n<p>Input the constant of integration as $C$.</p>", "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"]}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(4..9)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "m", "a", "s1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]</p>\n<p>$I=\\;$[[0]]</p>\n<p>Input numbers in your answer as integers or fractions and not as decimals.</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}], "resources": []}]}], "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}, {"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}]}