// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "\n  <p>Find the following integral.</p>\n  <p>\\[I = \\simplify[std]{Int({c}/((x +{a})*(x+{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>\n  ", "parts": [{"marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "steps": [{"marks": 0, "showCorrectAnswer": true, "prompt": "\n            \n            \n            <p>Use partial fractions in order to write:<br/>\\[\\simplify[std]{{c}/((x +{a})*(x+{b})) = A/(x+{a})+B/(x+{b})}\\]</p>\n            \n            \n            \n            <p>for suitable integers or fractions $A$ and $B$.</p>\n            \n            \n          ", "variableReplacements": [], "showFeedbackIcon": true, "customMarkingAlgorithm": "", "unitTests": [], "type": "information", "variableReplacementStrategy": "originalfirst", "scripts": {}, "extendBaseMarkingAlgorithm": true}], "showFeedbackIcon": true, "sortAnswers": false, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"expectedVariableNames": [], "marks": 3, "showCorrectAnswer": true, "checkVariableNames": false, "checkingAccuracy": 0.0001, "vsetRangePoints": 5, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "scripts": {}, "notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialCredit": 0, "strings": ["."], "showStrings": false}, "checkingType": "absdiff", "answerSimplification": "std", "failureRate": 1, "unitTests": [], "answer": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "variableReplacements": [], "showPreview": true, "type": "jme", "extendBaseMarkingAlgorithm": true, "vsetRange": [11, 12]}], "unitTests": [], "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, "type": "gapfill", "extendBaseMarkingAlgorithm": true}], "tags": [], "ungrouped_variables": ["s1", "c1", "c", "a", "b", "b1", "s3", "s2"], "name": "Simon's copy of Integration by partial fractions", "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "variables": {"s2": {"name": "s2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "c": {"name": "c", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "if(c1=2*a,c1+1,c1)"}, "b": {"name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "if(b1=a,b1+s3,b1)"}, "a": {"name": "a", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s1*random(1..9)"}, "s3": {"name": "s3", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}, "b1": {"name": "b1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s2*random(1..9)"}, "c1": {"name": "c1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(2..9)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}}, "advice": "<p>Using partial fractions we have 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 $\\displaystyle \\simplify[std]{((x +{a})*(x+{b}))}$ we obtain:</p>\n<p>$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}}\\Rightarrow \\simplify[std]{(A+B)*x+{b}A+{a}B={c}}$.&nbsp;</p>\n<p>One method to&nbsp;find A and B is by comparing coefficients:</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{{b}*A+{a}*B = {c}}$</p>\n<p>Coefficient of $x$: $ \\simplify[std]{A + B = 0}$ which gives $A = -B$</p>\n<p>Solving these simultaneous equations gives&nbsp;$\\displaystyle&nbsp;\\simplify[std]{A = {c}/{b-a}}$ 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 +{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<p>Or equivalently: &nbsp;$\\displaystyle I= \\simplify[std]{({c}/{b-a})*(ln((x+{a})/(x+{b})))+C}$</p>", "variable_groups": [], "metadata": {"description": "<p>Find $\\displaystyle\\int \\frac{a}{(x+b)(x+c)}\\;dx,\\;b \\neq c $.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}