// Numbas version: finer_feedback_settings
{"name": "Integral of improper polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sn*random(1..9)", "description": "", "name": "n"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sm*random(1..9)", "description": "", "name": "m"}, "sm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sm"}, "sp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sp"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sp*random(1..9)", "description": "", "name": "p"}, "sn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sn"}}, "ungrouped_variables": ["sp", "m", "n", "p", "sn", "sm"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Integral of improper polynomial fraction", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{n}/2*x^2 + {m} * x + {p} * arctan(x)+C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as fractions or integers and not decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n        <p>$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]</p>\n      <p>Input the arbitrary constant of integration as $C$.</p>\n        ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n    <p>Integrate the following function $f(x)$</p>\n    <p>\\[f(x)=\\simplify[std]{({n}x^3+{m}x^2+{n}x +{m+p})/(1+x^2)}\\]</p>\n    <p>Note that you can only enter inverse trigonometric functions as $\\arcsin(x),\\;\\;\\arccos(x),\\;\\;\\arctan(x)$</p>\n    <p>Input all numbers as fractions or integers and not decimals.</p>\n    ", "tags": ["arctan", "Calculus", "checked2015", "degree of a polynomial", "improper rational polynomials", "indefinite integration", "integration", "inverse trigonometric functions", "MAS1601", "polynomial division"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n        \t\t<p><strong>29/06/2012:</strong></p>\n        \t\t    <p><strong><br /></strong>Added tags. Tidied up display of prompt.</p>\n        \t\t    <p><strong>19/07/2012:</strong></p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>Checked calculation.</p>\n        \t\t    <p>Slight change to Advice, replaced \"long division\" by \"whatever way you like\" so not to prempt the method used by the student.</p>\n        \t\t    <p><strong>23/07/2012:</strong></p>\n        \t\t    <p></p>\n        \t\t    <p>Solution always requires arctan(x) and not arcsin(x) or arccos(x). Is this on purpose?</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t    <p>Question appears to be working correctly.</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{nx^3+mx^2+nx + p}{1+x^2}\\;dx$. Solution involves $\\arctan$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n    <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 division in whatever way you like, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{m}+{p}/(1+x^2)}}$.</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>$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{{n}x^2/2+{m}x+{p}arctan(x)} +C}$</p>\n    ", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}