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{"name": "Luis'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": [{"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 $\\displaystyle \\simplify[std]{1/((x +{a})*(x+{b}))}$ 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>", "preamble": {"css": "", "js": ""}, "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "pickQuestions": 0, "name": ""}], "name": "Luis's copy of Integration by partial fractions", "variable_groups": [], "type": "question", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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  ", "parts": [{"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      ", "steps": [{"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+{d})/((x +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]</p>\n            <p>for suitable integers or fractions $A$ and $B$.</p>\n          ", "type": "information", "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "scripts": {}, "gaps": [{"checkvariablenames": false, "answersimplification": "std", "scripts": {}, "type": "jme", "checkingtype": "absdiff", "notallowed": {"showStrings": false, "message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialCredit": 0, "strings": ["."]}, "expectedvariablenames": [], "marks": 3, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "vsetrange": [11, 12], "showpreview": true, "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "vsetrangepoints": 5}], "stepsPenalty": 1, "type": "gapfill", "showCorrectAnswer": true, "marks": 0}], "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "b1", "d1"], "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.&nbsp;</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>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>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": ["2 distinct linear factors", "Calculus", "MAS1601", "Steps", "checked2015", "completing the square", "constant of integration", "factorising a quadratic", "indefinite integration", "integration", "logarithms", "partial fractions", "two distinct linear factors"], "showQuestionGroupNames": false, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"c": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "c", "definition": "random(2..9)"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b", "definition": "if(b1=a,b1+s3,b1)"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b1", "definition": "s2*random(1..9)"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "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))"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s3", "definition": "random(1,-1)"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s2", "definition": "random(1,-1)"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "d1", "definition": "s3*random(1..9)"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a", "definition": "s1*random(1..9)"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s1", "definition": "random(1,-1)"}}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}