// Numbas version: finer_feedback_settings
{"name": "Maths Support: Integration using partial fractions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "<p>4 questions on using partial fractions to solve indefinite integrals.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 50.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": false, "showexpectedanswerswhen": "never", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["2 distinct linear factors", "Calculus", "Steps", "calculus", "compare coefficients", "identify coefficients", "integrals", "integration", "logarithms", "partial fractions", "steps", "two distinct linear factors"], "advice": "\n  <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 $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ we obtain:</p>\n  <p>$\\simplify[std]{A*(x+{b})+B*(x+{a}) = {c}}\\Rightarrow \\simplify[std]{(A+B)*x+{b}A+{a}B={c}}$.&nbsp;</p>\n  <p>We now identify coefficients on both sides of this equation.</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 +{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>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "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      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [11.0, 12.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "type": "jme"}], "steps": [{"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          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "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  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1..9)", "name": "a"}, "c": {"definition": "if(c1=2*a,c1+1,c1)", "name": "c"}, "b": {"definition": "if(b1=a,b1+s3,b1)", "name": "b"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b1": {"definition": "s2*random(1..9)", "name": "b1"}, "c1": {"definition": "random(2..9)", "name": "c1"}}, "metadata": {"notes": "\n    \t\t<p><strong>5/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Added decimal point as forbidden string.</p>\n    \t\t<p>Note the checking range is chosen so that the arguments of the log terms are always positive - could have used abs - might be better?</p>\n    \t\t<p>Improved display of Advice. Added alternative solution at end using log laws.</p>\n    \t\t<p>Added information about Show steps, also introduced penalty of 1 mark.</p>\n    \t\t<p>Added !noLeadingMinus to ruleset std for display purposes.</p>\n    \t\t", "description": "<p>Find $\\displaystyle\\int \\frac{a}{(x+b)(x+c)}\\;dx,\\;b \\neq c $.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["2 distinct linear factors", "Calculus", "Steps", "calculus", "compare coefficients", "identify coefficients", "integrals", "integration", "logarithms", "partial fractions", "steps", "two distinct linear factors"], "advice": "\n  <p>Using partial fractions we have to find $A$ and $B$ such that: <br/>\\[\\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b}))}\\;= \\simplify[std]{A/(x+{a})+B/(x+{b})}\\]<br/>Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ &nbsp; 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><strong>Identifying coefficients:</strong></p>\n    <p>Constant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$</p>\n    <p>Coefficent $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$</p>\n    <p>On solving these 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}))}\\; =\\simplify[std]{ ({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 +{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>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "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        \n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [11.0, 12.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "type": "jme"}], "steps": [{"prompt": "\n          <p>Use partial fractions in order to write:<br/>\\[\\simplify[std]{({c}*x+{d})/((x +{a})*(x+{b}))}\\; = \\simplify[std]{A/(x+{a})+B/(x+{b})}\\]</p>\n            <p>for suitable integers or fractions $A$ and $B$.</p>\n          ", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n    <p>Find the following integral.</p>\n    <p>\\[I = \\simplify[std]{Int(({c}*x+{d})/((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    \n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1..9)", "name": "a"}, "c": {"definition": "random(2..9)", "name": "c"}, "b": {"definition": "if(b1=a,b1+s3,b1)", "name": "b"}, "d": {"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))", "name": "d"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b1": {"definition": "s2*random(1..9)", "name": "b1"}, "d1": {"definition": "s3*random(1..9)", "name": "d1"}}, "metadata": {"notes": "\n    \t\t    \t\t<p><strong>5/08/2012:</strong></p>\n    \t\t    \t\t<p>Added tags.</p>\n    \t\t    \t\t<p>Added description.</p>\n    \t\t    \t\t<p>Added decimal point as forbidden string.</p>\n    \t\t    \t\t<p>Note the checking range is chosen so that the arguments of the log terms are always positive - could have used abs - might be better?</p>\n    \t\t    \t\t<p>Improved display of Advice.&nbsp;</p>\n    \t\t    \t\t<p>Added information about Show steps, also introduced penalty of 1 mark.</p>\n    \t\t    \t\t<p>Added !noLeadingMinus to ruleset std for display purposes.</p>\n    \t\t    \n    \t\t", "description": "<p>Find $\\displaystyle\\int \\frac{ax+b}{(x+c)(x+d)}\\;dx,\\;a\\neq 0,\\;c \\neq d $.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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"]}, "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", "type": "question"}, {"name": "Integration by partial fractions. (Video)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "tags": ["2 distinct linear factors", "Calculus", "Steps", "calculus", "completing the square", "constant of integration", "factorising a quadratic", "indefinite integration", "integrals", "integration", "logarithms", "partial fractions", "steps", "two distinct linear factors", "video"], "advice": "\n  <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 $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ 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><strong>Identifying coefficients:</strong></p>\n    <p>Constant term: $\\simplify[std]{{b}*A+{a}*B = {d}}$</p>\n    <p>Coefficent $x$: $ \\simplify[std]{A+B={c}}$ which gives $A =\\var{c} -B$</p>\n    <p>On solving these 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>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepspenalty": 1.0, "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>", "gaps": [{"notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [11.0, 12.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "type": "jme"}], "steps": [{"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>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "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  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(1..9)", "name": "a"}, "c": {"definition": "random(2..9)", "name": "c"}, "b": {"definition": "if(b1=a,b1+s3,b1)", "name": "b"}, "d": {"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))", "name": "d"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b1": {"definition": "s2*random(1..9)", "name": "b1"}, "d1": {"definition": "s3*random(1..9)", "name": "d1"}}, "metadata": {"notes": "\n                    \t\t    \t\t<p><strong>5/08/2012:</strong></p>\n                    \t\t    \t\t<p>Added tags.</p>\n                    \t\t    \t\t<p>Added description.</p>\n                    \t\t    \t\t<p>Added decimal point as forbidden string.</p>\n                    \t\t    \t\t<p>Note the checking range is chosen so that the arguments of the log terms are always positive - could have used abs - might be better?</p>\n                    \t\t    \t\t<p>Improved display of Advice.&nbsp;</p>\n                    \t\t    \t\t<p>Added information about Show steps, also introduced penalty of 1 mark.</p>\n                    \t\t    \t\t<p>Added !noLeadingMinus to ruleset std for display purposes.</p>\n                    \t\t    \n                    \t\t", "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"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "extensions": [], "custom_part_types": [], "resources": []}