// Numbas version: finer_feedback_settings
{"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": [{"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  ", "type": "question", "name": "Simon's copy of Integration by partial fractions", "variable_groups": [], "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0}], "progress": "ready", "extensions": [], "parts": [{"marks": 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        \n      ", "steps": [{"marks": 0.0, "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          "}], "type": "gapfill", "stepspenalty": 1.0, "gaps": [{"vsetrangepoints": 5.0, "marks": 3.0, "checkingtype": "absdiff", "answer": "({d-a*c}/{b-a})*ln(x+{a})+({d-b*c}/{a-b})*ln(x+{b})+C", "vsetrange": [11.0, 12.0], "checkingaccuracy": 0.0001, "type": "jme", "answersimplification": "std", "notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "strings": ["."], "partialcredit": 0.0, "showstrings": false}}]}], "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  ", "functions": {}, "variables": {"s2": {"name": "s2", "definition": "random(1,-1)"}, "c": {"name": "c", "definition": "random(2..9)"}, "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))"}, "s1": {"name": "s1", "definition": "random(1,-1)"}, "a": {"name": "a", "definition": "s1*random(1..9)"}, "s3": {"name": "s3", "definition": "random(1,-1)"}, "b": {"name": "b", "definition": "if(b1=a,b1+s3,b1)"}, "b1": {"name": "b1", "definition": "s2*random(1..9)"}, "d1": {"name": "d1", "definition": "s3*random(1..9)"}}, "tags": ["2 distinct linear factors", "Calculus", "Steps", "calculus", "compare coefficients", "identify coefficients", "integrals", "integration", "logarithms", "partial fractions", "steps", "two distinct linear factors"], "showQuestionGroupNames": false, "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", "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"}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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/"}]}