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{"name": "Integration 1 - Substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"templateType": "anything", "name": "b", "description": "", "definition": "repeat(random(-9..9 except 0),5)", "group": "Ungrouped variables"}, "m": {"templateType": "anything", "name": "m", "description": "", "definition": "repeat(random(4..9),5)", "group": "Ungrouped variables"}, "a": {"templateType": "anything", "name": "a", "description": "", "definition": "repeat(random(1..5),5)", "group": "Ungrouped variables"}}, "statement": "<p><strong>Integrate the following by substitution.</strong></p>\n<p>Don't forget the constant of integration ($C$).</p>", "name": "Integration 1 - Substitution", "parts": [{"marks": 0, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "<p>$I=\\simplify[std]{Int( x*({a[0]}x^2+{b[0]})^{m[0]},x)}$</p>\n<p></p>\n<p>Use $u=\\simplify[std]{{a[0]}x^2+{b[0]}}$ as your substitution.</p>\n<p>$\\frac{du}{dx}=$&nbsp;[[1]]</p>\n<p>$dx=$&nbsp;[[2]]</p>\n<p>Substituting back into the original equation for $dx$ and pulling out constants gives</p>\n<p>$I=$[[3]]$\\simplify[std]{Int(u^{m[0]},u)}$</p>\n<p>The next step is to integrate.</p>\n<p>$\\simplify{Int(u^{m[0]},u)}=$&nbsp;[[4]]</p>\n<p>Putting all of these results together, we get the final answer of:</p>\n<p>[[0]]</p>", "gaps": [{"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "notallowed": {"showStrings": false, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "strings": ["."], "partialCredit": 0}, "checkingAccuracy": 0.001, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "2", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "(({a[0]}*(x^2)+{b[0]})^({m[0]}+1))/(2*{a[0]}*({m[0]}+1))+C", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": 1, "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "2{a[0]}x", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "du", "value": ""}, {"name": "x", "value": ""}], "showFeedbackIcon": true, "marks": "1", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "du/(2{a[0]}x)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}, {"correctAnswerStyle": "plain", "marks": 1, "mustBeReduced": false, "customMarkingAlgorithm": "", "scripts": {}, "extendBaseMarkingAlgorithm": true, "customName": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "minValue": "1/(2{a[0]})", "mustBeReducedPC": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "numberentry", "maxValue": "1/(2{a[0]})", "allowFractions": true, "unitTests": [], "useCustomName": false, "correctAnswerFraction": true, "variableReplacements": []}, {"vsetRangePoints": 5, "customMarkingAlgorithm": "", "showPreview": true, "extendBaseMarkingAlgorithm": true, "customName": "", "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "checkingAccuracy": 0.001, "valuegenerators": [{"name": "u", "value": ""}], "showFeedbackIcon": true, "marks": "1", "checkVariableNames": false, "answerSimplification": "all", "vsetRange": [0, 1], "failureRate": 1, "answer": "u^({m[0]}+1)/({m[0]}+1)", "showCorrectAnswer": true, "scripts": {}, "type": "jme", "unitTests": [], "useCustomName": false, "variableReplacements": []}], "unitTests": [], "useCustomName": false, "variableReplacements": []}], "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "ungrouped_variables": ["a", "b", "m"], "advice": "<p>This problem is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a[0]} * (x ^ 2) + {b[0]}}$ then $du=\\simplify[std]{(2*{a[0]} * x)*dx }$<br/>Hence we can replace $xdx$ by $\\frac{1}{2*\\var{a[0]}}du$.</p>\n<p>Hence the integral becomes:</p>\n<p>\\[\\begin{eqnarray*} I&amp;=&amp;\\simplify[std]{Int((1/(2{a[0]}))u^{m[0]},u)}\\\\ &amp;=&amp;\\simplify[all]{(1/(2{a[0]}))u^{m[0]+1}/{m[0]+1}+C}\\\\ &amp;=&amp; \\simplify[all]{({a[0]} * (x ^ 2) + {b[0]})^{m[0]+1}/(2{a[0]}*({m[0]}+1))+C} \\end{eqnarray*}\\]</p>\n<p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ 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>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Step by step solving for integration by substitution</p>"}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}