// Numbas version: finer_feedback_settings
{"name": "Integration by partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Integration by partial fractions", "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"}], "extensions": [], "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": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}