// Numbas version: exam_results_page_options
{"name": "Simon'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": [{"tags": ["2 distinct linear factors", "Calculus", "Steps", "calculus", "completing the square", "constant of integration", "factorising a quadratic", "factorizing a quadratic", "indefinite integration", "integrals", "integration", "partial fractions", "steps", "two distinct linear factors"], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0, "questions": []}], "showQuestionGroupNames": false, "parts": [{"steps": [{"type": "information", "marks": 0.0, "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 +a)*(x+b)) = A/(x+a)+B/(x+b)}\\]</p>\n          <p>for suitable integers or fractions $A$ and $B$.</p>\n          "}], "stepspenalty": 1.0, "gaps": [{"checkingtype": "absdiff", "checkingaccuracy": 0.0001, "type": "jme", "marks": 3.0, "answer": "({c}/{b-a})*(ln(x+{a})-ln(x+{b}))+C", "vsetrangepoints": 5.0, "answersimplification": "std", "vsetrange": [11.0, 12.0], "notallowed": {"showstrings": false, "strings": ["."], "message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "partialcredit": 0.0}}], "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      ", "type": "gapfill", "marks": 0.0}], "type": "question", "progress": "ready", "statement": "\n  <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>\n  ", "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  ", "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", "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", "licence": "Creative Commons Attribution 4.0 International"}, "name": "Simon's copy of Integration by partial fractions", "functions": {}, "variables": {"a": {"name": "a", "definition": "s1*random(1..9)"}, "c": {"name": "c", "definition": "random(2..9)"}, "s2": {"name": "s2", "definition": "random(1,-1)"}, "s3": {"name": "s3", "definition": "random(1,-1)"}, "b1": {"name": "b1", "definition": "s2*random(1..9)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}, "b": {"name": "b", "definition": "if(b1=a,b1+s3,b1)"}}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variable_groups": [], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}