// Numbas version: exam_results_page_options
{"name": "Maria'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": [{"type": "question", "parts": [{"scripts": {}, "type": "gapfill", "gaps": [{"answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "type": "jme", "answersimplification": "std", "showCorrectAnswer": true, "vsetrangepoints": 5, "expectedvariablenames": [], "vsetrange": [11, 12], "checkingtype": "absdiff", "scripts": {}, "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "strings": ["."], "partialCredit": 0}, "marks": 3, "showpreview": true, "checkvariablenames": false, "checkingaccuracy": 0.0001}], "showCorrectAnswer": true, "stepsPenalty": 1, "marks": 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      ", "steps": [{"scripts": {}, "type": "information", "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          ", "showCorrectAnswer": true, "marks": 0}]}], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "functions": {}, "tags": ["2 distinct linear factors", "Calculus", "calculus", "checked2015", "compare coefficients", "identify coefficients", "integration", "logarithms", "MAS1601", "mas1601", "partial fractions", "Steps", "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  ", "showQuestionGroupNames": false, "name": "Maria's copy of Integration by partial fractions", "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": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle\\int \\frac{ax+b}{(x+c)(x+d)}\\;dx,\\;a\\neq 0,\\;c \\neq d $.</p>"}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "b1", "d1"], "variables": {"a": {"name": "a", "definition": "s1*random(1..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "d": {"name": "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))", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "b": {"name": "b", "definition": "if(b1=a,b1+s3,b1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s2": {"name": "s2", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s1": {"name": "s1", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "s3": {"name": "s3", "definition": "random(1,-1)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "c": {"name": "c", "definition": "random(2..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "b1": {"name": "b1", "definition": "s2*random(1..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "d1": {"name": "d1", "definition": "s3*random(1..9)", "templateType": "anything", "description": "", "group": "Ungrouped variables"}}, "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}