// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Simon's copy of Integration by partial fractions. (Video)", "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+{d})/(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    \n  ", "advice": "<p>First we factorise $\\simplify[std]{x^2+{a+b}*x+{a*b}=(x+{a})*(x+{b})}$. You can do this by spotting the factors or by completing the square.<br/>Next we use partial fractions to find $A$ and $B$ such that: <br/>\\[\\displaystyle \\simplify[std]{({c}*x+{d})/((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>$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}*x+{d}} \\Rightarrow \\simplify[std]{(A+B)*x+{b}*A+{a}*B={c}*x+{d}}$</p>\n<p>One way to find A and B is to compare coefficients:</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$</p>\n<p>Coefficient of $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$</p>\n<p>On solving these simultaneous equations we obtain $\\displaystyle&nbsp;\\simplify[std]{A = {d-a*c}/{b-a}}$ and $\\displaystyle&nbsp;\\simplify[std]{B={d-b*c}/{a-b}}$</p>\n<p>Which gives: \\[\\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b})) = ({d-a*c}/{b-a})*(1/(x+{a}) )+({d-b*c}/{a-b})*(1/(x+{b}))}\\]</p>\n<p>So \\[\\begin{eqnarray*} I &amp;=&amp; \\simplify[std]{Int(({c}*x+{d})/(x^2+{a+b}*x+{a*b}),x )}\\\\ &amp;=&amp;\\simplify[std]{Int(({c}*x+{d})/((x +{a})*(x+{b})),x )}\\\\ &amp;=&amp; \\simplify[std]{({d-a*c}/{b-a})*(Int(1/(x+{a}),x)) +({d-b*c}/{a-b})Int(1/(x+{b}),x)}\\\\ &amp;=&amp; \\simplify[std]{({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C} \\end{eqnarray*}\\]</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "name": "Simon's copy of Simon's copy of Integration by partial fractions. (Video)", "variable_groups": [], "ungrouped_variables": ["a", "d", "d1", "s2", "s3", "b1", "s1", "b", "c"], "parts": [{"marks": 0, "showCorrectAnswer": true, "prompt": "<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      <p>There is a video in Show steps which solves a similar example.</p>", "steps": [{"marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "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+{d})/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]</p>\n          <p>for suitable integers or fractions $A$ and $B$.</p>\n          <p>This video solves a similar, simpler example.</p>\n          <p><iframe src=\"https://player.vimeo.com/video/8183272\" frameborder=\"0\" width=\"640\" height=\"480\"></iframe></p>", "showFeedbackIcon": true, "unitTests": [], "type": "information", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "scripts": {}}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "sortAnswers": false, "scripts": {}, "gaps": [{"expectedVariableNames": [], "marks": 3, "showCorrectAnswer": true, "variableReplacements": [], "checkingType": "absdiff", "showFeedbackIcon": true, "unitTests": [], "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "scripts": {}, "notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialCredit": 0, "strings": ["."], "showStrings": false}, "checkingAccuracy": 0.001, "answerSimplification": "std", "failureRate": 1, "vsetRangePoints": 5, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "checkVariableNames": false, "showPreview": true, "type": "jme", "extendBaseMarkingAlgorithm": true, "vsetRange": [11, 12]}], "unitTests": [], "variableReplacements": [], "stepsPenalty": 1, "type": "gapfill", "extendBaseMarkingAlgorithm": true}], "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": "random(2..9)"}, "d": {"name": "d", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "if(d1=a*c,if(d1+1=b*c,d1+2,d1+1),if(d1=b*c,if(d1+1=a*c,d1+2,d1+1),d1))"}, "d1": {"name": "d1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s3*random(1..9)"}, "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)"}, "b": {"name": "b", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "if(b1=a,b1+s3,b1)"}, "b1": {"name": "b1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "s2*random(1..9)"}, "s1": {"name": "s1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1,-1)"}}, "tags": [], "functions": {}, "metadata": {"description": "<p>Factorise $x^2+cx+d$ into 2 distinct linear factors and then find $\\displaystyle \\int \\frac{ax+b}{x^2+cx+d}\\;dx,\\;a \\neq 0$ using partial fractions or otherwise.</p>\n                <p>Video in Show steps.</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/"}]}