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{"name": "MA1002 Integration by Substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "parts": [{"type": "gapfill", "marks": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"answer": "(({a[0]}*(x^2)+{b[0]})^({m[0]}+1))/(2*{a[0]}*({m[0]}+1))+C", "marks": "2", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "notallowed": {"partialCredit": 0, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "showStrings": false, "strings": ["."]}, "type": "jme", "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "variableReplacements": [], "vsetrange": [0, 1], "scripts": {}, "showCorrectAnswer": false}, {"marks": 1, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "showCorrectAnswer": false, "type": "jme", "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "variableReplacements": [], "vsetrange": [0, 1], "scripts": {}, "answer": "2{a[0]}x"}, {"marks": "1", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "showCorrectAnswer": false, "type": "jme", "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "variableReplacements": [], "vsetrange": [0, 1], "scripts": {}, "answer": "du/(2{a[0]}x)"}, {"marks": 1, "showFeedbackIcon": true, "correctAnswerStyle": "plain", "scripts": {}, "allowFractions": true, "correctAnswerFraction": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "minValue": "1/(2{a[0]})", "variableReplacements": [], "mustBeReducedPC": 0, "showCorrectAnswer": false, "maxValue": "1/(2{a[0]})", "notationStyles": ["plain", "en", "si-en"]}, {"marks": "1", "showFeedbackIcon": true, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "all", "showCorrectAnswer": false, "type": "jme", "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "variableReplacements": [], "vsetrange": [0, 1], "scripts": {}, "answer": "u^({m[0]}+1)/({m[0]}+1)"}], "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>", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true}], "ungrouped_variables": ["a", "b", "m"], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p><strong>Integrate the following by substitution.</strong></p>\n<p>Don't forget the constant of integration ($C$).</p>", "tags": [], "extensions": [], "name": "MA1002 Integration by Substitution", "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": {"description": "<p>Step by step solving for integration by substitution</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "variables": {"a": {"definition": "repeat(random(1..5),5)", "templateType": "anything", "description": "", "name": "a", "group": "Ungrouped variables"}, "m": {"definition": "repeat(random(4..9),5)", "templateType": "anything", "description": "", "name": "m", "group": "Ungrouped variables"}, "b": {"definition": "repeat(random(-9..9 except 0),5)", "templateType": "anything", "description": "", "name": "b", "group": "Ungrouped variables"}}, "type": "question", "contributors": [{"name": "Kieran Mulchrone", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1243/"}]}]}], "contributors": [{"name": "Kieran Mulchrone", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1243/"}]}