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{"name": "Integration with Partial Fractions 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration with Partial Fractions 4", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{cx+d}{(x+a)^2}$ using partial fractions.</p>", "licence": "None specified"}, "statement": "<p>Calculate the integral</p>\n<p>\\[ \\simplify{int(({c}x+{d})/(x+{a})^2,x)} .\\]</p>", "advice": "<p>In order to integrate the function \\[ \\simplify{({c}x+{d})/(x+{a})^2}, \\] we want to rewrite it in terms of its partial fractions.</p>\n<p>Since we have a repeated linear factor we want to set the function equal to the sum of 2 fractions with denominators $\\simplify{x+{a}}$ and $\\simplify{(x+{a})^2}$. The numerators will be constants, which we will call $A$ and $B$:</p>\n<p>\\[ \\simplify{({c}x+{d})/((x+{a})^2)} = \\simplify{A/(x+{a}) + B/(x+{a})^2}\\]</p>\n<p>To find the values of $A$ and $B$, we want to first multiply this equation by the denominator of the left-hand side. This gives</p>\n<p>\\[ \\simplify{{c}x+{d}=A(x+{a})+B}.\\]</p>\n<p></p>\n<p>By comparing the coefficients of the $x$-terms and the constant terms we can form a pair of simultaneous equations to find $A$ and $B$:</p>\n<p>\\[ \\begin{split} \\simplify{{c}x+{d}} &amp;\\,= \\simplify{A(x+{a})+B} \\\\ &amp;\\,= \\simplify{A*x+{a}A+B}. \\end{split} \\]</p>\n<p>\\[ \\begin{split}&amp;(x):\\quad \\var{c} &amp;\\,= \\simplify{A} \\\\ &amp;(c):\\quad \\var{d} &amp;\\,= \\simplify{{a}A+B} .\\end{split} \\]</p>\n<p>Hence,</p>\n<p>\\[A=\\var{c},\\,B=\\simplify{{d-a*c}}, \\]</p>\n<p>and</p>\n<p>\\[ \\simplify{({c}x+{d})/((x+{a})^2)} = \\simplify{{c}/(x+{a}) + {d-a*c}/(x+{a})^2}.\\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split} \\simplify{int(({c}x+{d})/((x+{a})^2),x)} &amp;\\,= \\simplify{int({Asol}/(x+{a}) + {Bsol}/(x+{a})^2,x)} \\\\ &amp;\\,=\\simplify[basic,fractionNumbers,noLeadingMinus]{{Asol} int(1/(x+{a}),x)+{Bsol} int((x+{a})^-2,x)} \\\\\\\\ &amp;\\,=\\simplify[all,fractionNumbers]{{Asol} ln(abs(x+{a}))-{Bsol}/(x+{a})+C}. \\end{split}\\]</p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except c*a)", "description": "", "templateType": "anything", "can_override": false}, "Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "c", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "d-a*c", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(c,d)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "d", "Asol", "Bsol"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "{Asol} ln(x+{a})-{Bsol}/(x+{a})+c", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "answer": "{Asol} ln(abs(x+{a}))-{Bsol}/(x+{a})+C", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}