// Numbas version: exam_results_page_options
{"name": "Andrew's copy of CF Maths In class test three question 8 Integration by partial fractions with limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "b1", "d1"], "name": "Andrew's copy of CF Maths In class test three question 8 Integration by partial fractions with limits", "tags": ["calculus", "Calculus", "compare coefficients", "integrals", "integration", "logarithms", "partial fractions", "Steps", "steps"], "preamble": {"css": "", "js": ""}, "advice": "<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})} \\end{eqnarray*}\\]</p>\n<p></p>\n<p>For the last step substitute the upper limt and lower limit separately to the integral, and the value&nbsp;from upper limt take away from the lower one to give the final answer.</p>\n<p>\\[\\simplify[std]{({d-a*c}/{b-a})*ln(12+{a})+({d-b*c}/{a-b})*ln(12+{b})} \\] <strong>take away</strong> \\[\\simplify[std]{({d-a*c}/{b-a})*ln(10+{a})+({d-b*c}/{a-b})*ln(10+{b})} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>First, input the sum of partial fraction</p>\n<p>[[0]]</p>\n<p></p>\n<p>Second,&nbsp;input the integral without the limit</p>\n<p>$I=\\;$[[1]]</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p><strong>Do not</strong> need to input $C$, the constant of integration.</p>\n<p></p>\n<p>The last, substitute the limits to the&nbsp;integral and calculate it.</p>\n<p>[[2]], accurate to 2 decimal places.</p>\n<p>Click on Show steps for help if you need it. You will lose 1 mark if you do so.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({d-a*c}/(({b-a})*(x+{a})))+({d-b*c}/(({a-b})*(x+{b})))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.0001, "vsetrange": [11, 12], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "(({d-a*c}/{b-a})*ln(12+{a})+({d-b*c}/{a-b})*ln(12+{b}))-(({d-a*c}/{b-a})*ln(10+{a})+({d-b*c}/{a-b})*ln(10+{b}))", "strictPrecision": true, "minValue": "(({d-a*c}/{b-a})*ln(12+{a})+({d-b*c}/{a-b})*ln(12+{b}))-(({d-a*c}/{b-a})*ln(10+{a})+({d-b*c}/{a-b})*ln(10+{b}))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Find the following integral.</p>\n<p>\\[I = \\simplify[std]{defint(({c}*x+{d})/((x +{a})*(x+{b})),x,10,12 )}\\]</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "if(b1=a,b1+s3,b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "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", "group": "Ungrouped variables", "name": "d", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "b1": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "d1": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}}, "metadata": {"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"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}]}]}], "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}]}