// Numbas version: exam_results_page_options
{"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": {"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"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sn*random(1..9)", "description": "", "name": "n"}}, "ungrouped_variables": ["p", "sp", "sn", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Integral of improper polynomial fraction", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "((({n} * (x ^ 2)) / 2) + ({p} * Arctan(x))+C)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>&nbsp;Input all numbers as fractions or integers and not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<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>", "showCorrectAnswer": true, "marks": 0}], "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>Note that you can only enter inverse trigonometric functions as $\\arcsin(x),\\;\\;\\arccos(x),\\;\\;\\arctan(x)$.</p>\n  <p>&nbsp;</p>", "tags": ["arctan", "Calculus", "checked2015", "degree of a polynomial", "improper rational polynomials", "indefinite integration", "integration", "integration of standard functions", "integration using trigonometric identities", "inverse trigonometric functions", "MAS1601", "polynomial division", "trigonometric identities"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>28/06/2012:</strong></p>\n                <p><strong><br /></strong>Added tags.</p>\n                <p>Improved display of question prompt.</p>\n                <p>Changed instructions for inputting integration constant</p>\n                <p>Added decimal point . as forbidden string to stop decimal input (is this necessary?)</p>\n                <p><strong>18/07/2012:</strong></p>\n                <p>Added description.</p>\n                <p><strong>23/07/2012:</strong></p>\n                <p>Added tags.</p>\n                <p>Solution always requires arctan(x) and not arcsin(x) or arccos(x). Is this on purpose?</p>\n                <p>Question appears to be working correctly.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int\\frac{ax^3+ax+b}{1+x^2}\\;dx$. Enter the constant of integration as $C$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "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/"}]}