// Numbas version: exam_results_page_options
{"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "advice": "\n  \n  \n  <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  \n  \n  \n  <p>Hence the integral becomes:</p>\n  \n  \n  \n  <p>\\[\\begin{eqnarray*} I&#38;=&#38;\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n  \n  &#38;=&#38;\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n  \n  &#38;=&#38; \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n  \n  \\end{eqnarray*}\\]</p>\n  \n  \n  \n  <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  \n  ", "name": "Integration by substitution", "preamble": {"js": "", "css": ""}, "variables": {"a": {"definition": "random(1..5)", "name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "m": {"definition": "random(4..9)", "name": "m", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "s1": {"definition": "random(1,-1)", "name": "s1", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "b": {"definition": "s1*random(1..9)", "name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>"}, "parts": [{"sortAnswers": false, "gaps": [{"showFeedbackIcon": true, "checkVariableNames": false, "type": "jme", "failureRate": 1, "unitTests": [], "checkingType": "absdiff", "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}], "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "vsetRange": [0, 1], "checkingAccuracy": 0.001, "notallowed": {"message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "strings": ["."], "partialCredit": 0, "showStrings": false}, "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "scripts": {}, "answerSimplification": "std", "showPreview": true, "customName": "", "customMarkingAlgorithm": "", "marks": 3, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "showFeedbackIcon": true, "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>Click on Show steps to get further help. You will lose 1 mark if you do so.</p>", "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true, "customName": "", "unitTests": [], "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "stepsPenalty": 1, "useCustomName": false, "steps": [{"showFeedbackIcon": true, "prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$</p>", "customMarkingAlgorithm": "", "type": "information", "showCorrectAnswer": true, "customName": "", "unitTests": [], "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false, "extendBaseMarkingAlgorithm": true}], "extendBaseMarkingAlgorithm": true}], "ungrouped_variables": ["a", "s1", "b", "m"], "statement": "<p>Find the following integral using substitution.</p>\n<p>Don't forget&nbsp;$C$!</p>", "tags": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}