// Numbas version: exam_results_page_options
{"name": "Indefinite integral by substitution 2 (custom feedback)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"m": {"group": "Ungrouped variables", "name": "m", "templateType": "anything", "definition": "random(3..9)", "description": ""}, "a": {"group": "Ungrouped variables", "name": "a", "templateType": "anything", "definition": "random(1..9)", "description": ""}, "b": {"group": "Ungrouped variables", "name": "b", "templateType": "anything", "definition": "s1*random(1..9)", "description": ""}, "s1": {"group": "Ungrouped variables", "name": "s1", "templateType": "anything", "definition": "random(1,-1)", "description": ""}}, "advice": "\n\t    \n\t    \n\t    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a}+{b}cos(x)}$ then $du=\\simplify[std]{({-b}*sin(x))*dx }$<br/>Hence we can replace $\\sin(x)\\;dx$ by $\\frac{1}{\\var{-b}}\\;du$.</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/{-b})u^{m},u)}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n\t    \n\t    \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t  \n\t", "name": "Indefinite integral by substitution 2 (custom feedback)", "variable_groups": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": [], "statement": "\n\t  <p>Find the following integral.</p>\n\t    <p>Input the constant of integration as $C$.</p>\n\t    <p>Input all numbers as integers or fractions not as decimals.</p>\n\t  \n\t", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$</p>"}, "parts": [{"stepsPenalty": 1, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "sortAnswers": false, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "gaps": [{"type": "jme", "showPreview": true, "marks": 3, "valuegenerators": [{"value": "", "name": "c"}, {"value": "", "name": "x"}], "unitTests": [], "answer": "-({a}+{b}*cos(x))^{m+1}/{b*(m+1)}+C", "customMarkingAlgorithm": "malrules:\n  [\n    [\"-({a}+{b}*cos(x))^{m+1}/{b*(m+1)}\", \"Almost there! Did you forget to include the integration constant?\"],\n    [\"u^{m+1}/{b*(m+1)}+C\",\"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\"],\n    [\"u^{m+1}/{b*(m+1)}\",\"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\"],\n    [\"-u^{m+1}/{b*(m+1)}+C\",\"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\"],\n    [\"-u^{m+1}/{b*(m+1)}\",\"Don't forget to fill back in for $u$. You must give your answer in terms of the original variable i.e. in terms of $x$.\"],\n    [\"-1/{b}*({a}+{b}*cos(x))^{m}+C\",\"You haven't actually integrated here. You simply subbed in for $\\\\var{a} + \\\\var{b} \\\\cos x$ and then subbed back again.\"],\n    [\"-1/{b}*({a}+{b}*cos(x))^{m}\",\"You haven't actually integrated here. You simply subbed in for $\\\\var{a} + \\\\var{b} \\\\cos x$ and then subbed back again.\"],\n    [\"-1/{b}*u^{m}+C\",\"You haven't actually integrated here. You simply subbed in for $\\\\var{a} + \\\\var{b} \\\\cos x$. Also, once you have integrated, make sure you sub back in for $u$ - give your answer in terms of the original variable (i.e. in terms of $x$).\"],\n    [\"-1/{b}*u^{m}\",\"You haven't actually integrated here. You simply subbed in for $\\\\var{a} + \\\\var{b} \\\\cos x$. Also, once you have integrated, make sure you sub back in for $u$ - give your answer in terms of the original variable (i.e. in terms of $x$).\"],\n    [\"-cos(x)*({a}*x+{b}*sin(x))^{m}+C\",\"You cannot simply integrate each part of a product individually. Look closely at the integrand (what you were asked to integrate). Is one part of the integrand equal to the derivative (or a multiple of the derivative) of another part? If so, try substitution.\"], \n    [\"-cos(x)*({a}*x+{b}*sin(x))^{m}\",\"You cannot simply integrate each part of a product individually. Look closely at the integrand (what you were asked to integrate). Is one part of the integrand equal to the derivative (or a multiple of the derivative) of another part? If so, try substitution.\"], \n    [\"({a}+{b}*cos(x))^{m+1}/{b*(m+1)}+C\",\"Almost there. Look closely at what you let $u=$ and how you differentiated that. Double check the relevant differentiation rule.\"],\n    [\"({a}+{b}*cos(x))^{m+1}/{b*(m+1)}\",\"Almost there. Look closely at what you let $u=$ and how you differentiated that. Double check the relevant differentiation rule.\"],\n    [\"-({a}+{b}*cos(x))/{b*(m+1)}+C\", \"Be careful when subbing back in for $u$. Did you do this correctly? Did you include the power on the bracket?\"],\n    [\"-({a}+{b}*cos(x))/{b*(m+1)}\", \"Be careful when subbing back in for $u$. Did you do this correctly? Did you include the power on the bracket?\"],\n    [\"-({a}+{b}*cos(x))^{m+1}/{m+1}+C\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"],\n    [\"-({a}+{b}*cos(x))^{m+1}/{m+1}\", \"Double check the relationship between $du$ and $dx$. Did you sub in correctly here?\"]\n]\n\n\nparsed_malrules: \n    map(\n        [\"expr\":parse(x[0]),\"feedback\":x[1]],\n        x,\n        malrules\n    )\n\nagree_malrules (Do the student's answer and the expected answer agree on each of the sets of variable values?):\n    map(\n        len(filter(not x ,x,map(\n            try(\n                resultsEqual(unset(question_definitions,eval(studentexpr,vars)),unset(question_definitions,eval(malrule[\"expr\"],vars)),settings[\"checkingType\"],settings[\"checkingAccuracy\"]),\n                message,\n                false\n            ),\n            vars,\n            vset\n        )))<settings[\"failureRate\"],\n        malrule,\n        parsed_malrules\n    )\n\nmatched_malrules:\n  filter(x[0],x,zip(agree_malrules,parsed_malrules))\n  \nmalrule_feedback:\n    assert(numericallyCorrect,\n        assert(len(matched_malrules)=0,\n            set_credit(0,matched_malrules[0][1][\"feedback\"])\n        )\n    )\n\nmark:\n    apply(studentExpr);\n    apply(unexpectedVariables);\n    apply(numericallyCorrect);\n    apply(malrule_feedback);\n    apply(failMinLength);\n    apply(failMaxLength);\n    apply(forbiddenStringsPenalty);\n    apply(requiredStringsPenalty)", "checkingAccuracy": 0.001, "adaptiveMarkingPenalty": 0, "vsetRangePoints": 5, "variableReplacements": [], "notallowed": {"showStrings": false, "partialCredit": 0, "strings": ["."], "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions.</p>"}, "answerSimplification": "std", "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "checkingType": "absdiff", "showFeedbackIcon": true, "failureRate": 1, "customName": "", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "checkVariableNames": false, "scripts": {}}], "prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]</p>\n\t\t\t        <p>Input all numbers as integers or fractions.</p>\n\t\t\t        <p>$I=\\;$[[0]]</p>\n\t\t\t        <p>Input the constant of integration as $C$.</p>\n\t\t\t        <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n\t\t\t      \n\t\t\t", "steps": [{"type": "information", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "useCustomName": false, "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "prompt": "<p>Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$</p>", "customMarkingAlgorithm": "", "customName": "", "scripts": {}, "adaptiveMarkingPenalty": 0}], "customMarkingAlgorithm": "", "customName": "", "scripts": {}, "adaptiveMarkingPenalty": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "preamble": {"css": "", "js": ""}, "functions": {}, "ungrouped_variables": ["m", "a", "b", "s1"], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}]}