// Numbas version: exam_results_page_options
{"name": "Integration with Partial Fractions 2", "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", "nb", "a1", "a2", "s1", "a_", "c_", "b_", "d_", "nb_", "a1_", "a2_", "s1_", "i1", "i0"], "name": "Integration with Partial Fractions 2", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "preamble": {"css": "", "js": ""}, "advice": "<p>a)</p>\n<p>We use partial fractions to find $A$ and $B$ such that:&nbsp;<br/>\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \\;\\;\\;=\\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}\\]</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})({a2}x+{d})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$\\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \\Rightarrow \\simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{ {d}*A+{b}*B={a*d+c*b}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a2}A+{a1}B&nbsp;= {a*a2+c*a1}&nbsp;}$&nbsp;</p>\n<p>On solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$</p>\n<p>Which gives:\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\\;\\;= \\simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}\\]</p>\n<p><strong>Integration:</strong></p>\n<p>For the integration, we use our answer for the partial fractions to help use and then recall that $\\int \\frac{1}{x+a}\\mathrm{d}x = \\ln(x+a)$</p>\n<p>Hence we apply this to our problem to find $\\int \\simplify{({a*a2 + &nbsp;c*a1} * x + {a * d + &nbsp;c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}&nbsp;\\mathrm{d}x = \\simplify{({a}/{a1})*ln({a1}x+{b}) + ({c}/{a2})*ln({a2}x+{d})}$</p>\n<p>b)</p>\n<p>We use partial fractions to find $A$ and $B$ such that:&nbsp;<br/>\\[ \\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))} \\;\\;\\;=\\simplify[std]{ A/({a1_}x+{b_})+B/({a2_}x+{d_})}\\]</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1_}x+{b_})({a2_}x+{d_})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$\\simplify[std]{A*({a2_}x+{d_})+B*({a1_}x+{b_}) = {a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}} \\Rightarrow \\simplify[std]{({a2_}A+{a1_}B)*x+{d_}*A+{b_}*B={a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{ {d_}*A+{b_}*B={a_*d_+c_*b_}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a2_}A+{a1_}B&nbsp;= {a_*a2_+c_*a1_}&nbsp;}$&nbsp;</p>\n<p>On solving these equations we obtain $A = \\var{a_}$ and $B=\\var{c_}$</p>\n<p>Which gives:\\[ \\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))}\\;\\;= \\simplify[std]{{a_}/({a1_}x+{b_})+{c_}/({a2_}x+{d_})}\\]</p>\n<p><strong>Integration:</strong></p>\n<p>For the integration, we use our answer for the partial fractions to help use and then recall that $\\int \\frac{1}{x+a}\\mathrm{d}x = \\ln(x+a)$</p>\n<p>Hence we apply this to our problem to find $\\int \\simplify{({a_*a2_ + &nbsp;c_*a1_} * x + {a_ * d_ + &nbsp;c_ * b_})/ (({a1_}*x + {b_}) * ({a2_}*x + {d_}))}&nbsp;\\mathrm{d}x = \\simplify{({a_}/{a1_})*ln({a1_}x+{b_}) + ({c_}/{a2_})*ln({a2_}x+{d_})} $</p>\n<p>We can then evaluate&nbsp;$\\int^\\var{i1}_\\var{i0} \\simplify{({a_*a2_ + &nbsp;c_*a1_} * x + {a_ * d_ + &nbsp;c_ * b_})/ (({a1_}*x + {b_}) * ({a2_}*x + {d_}))}&nbsp;\\mathrm{d}x$ by simply subbing in the appropriate limits to our answer above</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Split \\[\\simplify{({a*a2 + &nbsp;c*a1} * x + {a * d + &nbsp;c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>Hence evaluate $\\int \\simplify{({a*a2 + &nbsp;c*a1} * x + {a * d + &nbsp;c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}&nbsp;\\mathrm{d}x =$[[1]]</p>\n<p>(You may assume that the constant on integration is 0)</p>\n<p></p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "strings": [")(", ")*("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 1e-05, "vsetrange": [10, 11], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d}))", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}/{a1})*ln({a1}x+{b}) + ({c}/{a2})*ln({a2}x+{d})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Split \\[\\simplify{({a_*a2_ + &nbsp;c_*a1_} * x + {a_ * d_ + &nbsp;c_ * b_})/ (({a1_}*x + {b_}) * ({a2_}*x + {d_}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]]</p>\n<p>Hence evaluate $\\int^\\var{i1}_\\var{i0} \\simplify{({a_*a2_ + &nbsp;c_*a1_} * x + {a_ * d_ + &nbsp;c_ * b_})/ (({a1_}*x + {b_}) * ({a2_}*x + {d_}))} \\mathrm{d}x =$[[1]]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "strings": [")(", ")*("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 1e-05, "vsetrange": [10, 11], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a_} / ({a1_}*x + {b_}) + ({c_} / ({a2_}*x + {d_}))", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(({a_}/{a1_})*ln({a1_}{i1} + {b_}) + ({c_}/{a2_})*ln({a2_}{i1} + {d_}))-(({a_}/{a1_})*ln({a1_}{i0} + {b_}) + ({c_}/{a2_})*ln({a2_}{i0} + {d_}))", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "a1", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "-a2", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "nb_": {"definition": "if(c_<0,'taking away','adding')", "templateType": "anything", "group": "Ungrouped variables", "name": "nb_", "description": ""}, "d": {"definition": "random(1..9 except [0,round(b*a2/a1)])", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "i1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}, "i0": {"definition": "floor(i1/2)", "templateType": "anything", "group": "Ungrouped variables", "name": "i0", "description": ""}, "nb": {"definition": "if(c<0,'taking away','adding')", "templateType": "anything", "group": "Ungrouped variables", "name": "nb", "description": ""}, "a2_": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a2_", "description": ""}, "a1": {"definition": " 1", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "d_": {"definition": "random(1..9 except [0,round(b_*a2_/a1_)])", "templateType": "anything", "group": "Ungrouped variables", "name": "d_", "description": ""}, "a2": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "c_": {"definition": "-a2_", "templateType": "anything", "group": "Ungrouped variables", "name": "c_", "description": ""}, "s1_": {"definition": "if(c_<0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1_", "description": ""}, "b_": {"definition": "random(1..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b_", "description": ""}, "a1_": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a1_", "description": ""}, "a_": {"definition": "a1_", "templateType": "anything", "group": "Ungrouped variables", "name": "a_", "description": ""}, "s1": {"definition": "if(c<0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}}, "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>Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.</p>\n        \t\t                    \t\t                                        <p><strong>12/08/2012:</strong></p>\n        \t\t                    \t\t                                        <p>Back to one input of a fraction and trapped input in Forbidden Strings.</p>\n        \t\t                    \t\t                                        <p>Used the except feature of ranges to get non-degenerate examples.</p>\n        \t\t                    \t\t                                        <p>Checked calculation.OK.</p>\n        \t\t                    \t\t                                        <p>Improved display in content areas.</p>\n        \t\t                    \n        \t\t", "description": "<p>Split $\\displaystyle \\frac{b}{(cx + d)(px+q)}$ into partial fractions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}