// Numbas version: exam_results_page_options
{"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": [{"tags": [], "parts": [{"unitTests": [], "steps": [{"showCorrectAnswer": true, "unitTests": [], "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "information", "marks": 0, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "\n          <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>\n          "}], "gaps": [{"unitTests": [], "checkingType": "absdiff", "answer": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "scripts": {}, "checkingAccuracy": 0.0001, "type": "jme", "showFeedbackIcon": true, "vsetRange": [11, 12], "showPreview": true, "expectedVariableNames": [], "showCorrectAnswer": true, "answerSimplification": "std", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "checkVariableNames": false, "marks": 3, "variableReplacements": [], "notallowed": {"strings": ["."], "message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialCredit": 0, "showStrings": false}, "variableReplacementStrategy": "originalfirst", "failureRate": 1}], "scripts": {}, "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "stepsPenalty": 1, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "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      "}], "statement": "\n  <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>\n  ", "advice": "<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 $\\displaystyle \\simplify[std]{((x +{a})*(x+{b}))}$ 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>One method to 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, we&nbsp;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>", "ungrouped_variables": ["s2", "c", "b1", "s3", "a", "s1", "b"], "preamble": {"css": "", "js": ""}, "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"}, "name": "Simon's copy of Integration by partial fractions", "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"a": {"templateType": "anything", "name": "a", "group": "Ungrouped variables", "description": "", "definition": "s1*random(1..9)"}, "b": {"templateType": "anything", "name": "b", "group": "Ungrouped variables", "description": "", "definition": "if(b1=a,b1+s3,b1)"}, "s2": {"templateType": "anything", "name": "s2", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "s3": {"templateType": "anything", "name": "s3", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "b1": {"templateType": "anything", "name": "b1", "group": "Ungrouped variables", "description": "", "definition": "s2*random(1..9)"}, "s1": {"templateType": "anything", "name": "s1", "group": "Ungrouped variables", "description": "", "definition": "random(1,-1)"}, "c": {"templateType": "anything", "name": "c", "group": "Ungrouped variables", "description": "", "definition": "random(2..9)"}}, "extensions": [], "variable_groups": [], "functions": {}, "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/"}]}