// Numbas version: exam_results_page_options
{"name": "Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by partial fractions", "tags": ["2 distinct linear factors", "calculus", "Calculus", "completing the square", "constant of integration", "factorising a quadratic", "factorizing a quadratic", "indefinite integration", "integrals", "integration", "partial fractions", "Steps", "steps", "two distinct linear factors"], "metadata": {"description": "\n    \t\t<p>Factorise $x^2+bx+c$ into 2 distinct linear factors and then find $\\displaystyle \\int \\frac{a}{x^2+bx+c }\\;dx$ using partial fractions or otherwise.</p>\n    \t\t<p>&nbsp;</p>\n    \t\t", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the following integral.</p>\n<p>\\[I = \\simplify[std]{Int({c}/(x^2+{a+b}*x+{a*b}),x )}\\]</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p>Input the constant of integration as $C$.</p>", "advice": "\n  <p>First we factorise $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$.</p>\n  <p>You can do this by spotting the factors or by completing the square.</p>\n  <p>Next we use partial fractions to find $A$ and $B$ such that: <br/>\\[\\simplify[std]{{c}/((x +{a})*(x+{b})) = A/(x+{a})+B/(x+{b})}\\]<br/>Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ we obtain:</p>\n  <p>$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}}\\Rightarrow \\simplify[std]{(A+B)x+{b}A+{a}B={c}}$</p>\n  <p><strong>Identifying coefficients:</strong></p>\n  <p>Constant term: $\\simplify[std]{{b}*A+{a}*B = {c}}$</p>\n  <p>Coefficent $x$: $ \\simplify[std]{A + B = 0}$ which gives $A = -B$.</p>\n  <p>Hence we obtain $\\displaystyle&nbsp;\\simplify[std]{A = {c}/{b-a}}$ and $\\displaystyle&nbsp;\\simplify[std]{B={-c}/{b-a}}$</p>\n  <p>Which gives: \\[\\simplify[std]{{c}/((x +{a})*(x+{b})) = ({c}/{b-a})*(1/(x+{a}) -1/(x+{b}))}\\]</p>\n  <p>So \\[\\begin{eqnarray*} I &amp;=&amp; \\simplify[std]{Int({c}/(x^2+{a+b}*x+{a*b}),x )}\\\\ &amp;=&amp; \\simplify[std]{Int({c}/((x +{a})*(x+{b})),x )}\\\\ &amp;=&amp; \\simplify[std]{({c}/{b-a})*(Int(1/(x+{a}),x) -Int(1/(x+{b}),x))}\\\\ &amp;=&amp; \\simplify[std]{({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C} \\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "if(b1=a,b1+s3,b1)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "b1"], "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": "\n      <p>$I=\\;$[[0]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n        <p>Input the constant of integration as $C$.</p>\n        <p>Click on Show steps for help if you need it. You will lose 1 mark if you do so.</p>\n      ", "stepsPenalty": 1, "steps": [{"type": "information", "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>First of all factorise the denominator.</p>\n<p>You have to find $a$ and $b$ such that $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+a)*(x+b)}$.</p>\n<p>Then use partial fractions to write:<br/>\\[\\simplify[std]{{c}/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]</p>\n<p>for suitable integers or fractions $A$ and $B$.</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": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.0001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [11, 12], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers and not decimals.</p>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}]}