// Numbas version: exam_results_page_options
{"name": "Simon's copy of Integrate algebraic fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Find $\\displaystyle \\int\\frac{ax^3+ax+b}{1+x^2}\\;dx$. Enter the constant of integration as $C$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variable_groups": [], "variables": {"sn": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sn"}, "sp": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sp"}, "n": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "sn*random(1..9)", "name": "n"}, "p": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "sp*random(1..9)", "name": "p"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "sp", "sn", "n"], "statement": "<p>Integrate the following function $f(x)$</p>\n<p>\\[f(x)=\\simplify[std]{({n}x^3+{n}x+{p})/(1+x^2)}\\]</p>\n<p></p>\n<p>You may need to use the standard result $\\int \\frac{1}{1+x^2}dx = \\arctan{x}+C$</p>\n<p>&nbsp;</p>", "advice": "<p>Since the degree of the numerator of $f(x)$ is greater than the denominator, $f(x)$ is improper.</p>\n<p>First, perform a polynomial long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{p}/(1+x^2)}}$.</p>\n<p>Each term of this expression can then be integrated using standard functions (to within the arbitrary constant) to give:</p>\n<p>$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{{n}x^2/2+{p}arctan(x)} +C}$</p>", "tags": [], "parts": [{"scripts": {}, "gaps": [{"scripts": {}, "failureRate": 1, "showPreview": true, "answer": "((({n} * (x ^ 2)) / 2) + ({p} * Arctan(x))+C)", "showCorrectAnswer": true, "checkVariableNames": false, "type": "jme", "answerSimplification": "std", "customMarkingAlgorithm": "", "variableReplacements": [], "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "expectedVariableNames": [], "extendBaseMarkingAlgorithm": true, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "vsetRange": [0, 1], "notallowed": {"showStrings": false, "message": "<p>&nbsp;Input all numbers as fractions or integers and not as decimals.</p>", "partialCredit": 0, "strings": ["."]}, "unitTests": [], "marks": 1}], "sortAnswers": false, "showCorrectAnswer": true, "type": "gapfill", "customMarkingAlgorithm": "", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "prompt": "\n      <p>$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]</p>\n        <p>You must input the arbitrary constant of integration as $C$.</p>\n        <p>Also input all numbers as fractions or integers and not as decimals.</p>\n      ", "extendBaseMarkingAlgorithm": true, "unitTests": [], "marks": 0}], "extensions": [], "functions": {}, "preamble": {"css": "", "js": ""}, "name": "Simon's copy of Integrate algebraic fraction", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}