// Numbas version: exam_results_page_options
{"name": "Indefinite integral of 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": "Indefinite integral of polynomial fraction", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{n}/2*x^2 + {m} * x + {p}/2 * ln(1+x^2)+C", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]</p>\n        <p>Input the arbitrary constant of integration as $C$.</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Integrate the following function $f(x)$</p>\n  <p>\\[f(x)=\\simplify[std]{({n}x^3+{m}x^2+{n+p}x +{m})/(1+x^2)}\\]</p>\n  <p>Note that if you need to enter the absolute value in your answer, e.g. $|x|$, then you should not use the vertical bar on the keyboard.</p>\n  <p>Instead you must use the abs() function, i.e. abs(x).</p>", "tags": ["Calculus", "checked2015", "degree of polynomial", "indefinite integration", "integration", "logarithm", "logs", "long division of polynomials", "MAS1601", "polynomial division", "polynomials"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>2/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Checked calculation.</p>\n    <p><strong>19/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p>&nbsp;</p>\n    <p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{nx^3+mx^2+px +m}{x^2+1} \\;dx$</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n  \n  \n  <p>Since the degree of the numerator of $f(x)$ is greater than the denominator,  $f(x)$ is improper.</p>\n  \n  \n  \n  <p>First, perform a long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{m}+({p}x)/(1+x^2)}}$.</p>\n  \n  \n  \n  <p>Each term of this expression can then be integrated using standard functions (to within the arbitrary constant) to give:</p>\n  \n  \n  \n  <p>$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{{n}x^2/2+{m}x+{p}/2*ln(1+x^2)} +C}$</p>\n  \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/"}]}