// Numbas version: finer_feedback_settings
{"name": "Indefinite 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": "Indefinite integral of improper polynomial fraction", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "(({ - n} * (x ^ 2)) / 2) + ({p} / 2) * ln(abs((1 + x) / (1 -x)))+C", "vsetrange": [0.25, 0.75], "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": 3, "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>Input all numbers as fractions or integers and not 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>Remember that the correct integration of expressions such as $\\displaystyle \\frac{1}{x-a}$ is:</p>\n  <p>\\[\\int \\frac{1}{x-a}\\;dx=\\ln(|x-a|)+C\\]</p>", "tags": ["absolute value", "Calculus", "checked2015", "improper rational polynomials", "indefinite integral", "indefinite integration", "integration", "integration of rational polynomials", "long division of polynomials", "MAS1601", "natural logarithm", "natural logs", "partial fractions", "polynomial division", "rational polynomials"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>28/06/2012:</strong></p>\n    <p>Added tags. Changed prompt slightly. Calculation checks.</p>\n    <p>Checking range is 2 to 3, giving a negative argument for ln without taking the absolute value, hence the reason for asking for absolute value being used. But in fact the user can just wrap abs around the $1+x$ and not the $1-x$. So is redundant!!</p>\n    <p>To be changed!</p>\n    <p><strong>29/06/2012:</strong></p>\n    <p>Got rid of necessity to have abs by changing checking range to 0.25 to 0.75. However, still including the homily about using abs as good practice. Cannot use in Show steps for now as not available (issue reported) so is in the statement and also in Advice.</p>\n    <p><strong>10/07/2012:</strong></p>\n    <p>Included message in the prompt about using fractions or integers, not decimals. Included decimal point in forbidden strings.</p>\n    <p><strong>18/07/2012:</strong></p>\n    <p>Added description.</p>\n    <p>23/07/2012:</p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle\\int \\frac{ax^3-ax+b}{1-x^2}\\;dx$. Input 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 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>$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{ (({ - n} * (x ^ 2)) / 2) + ({p} / 2) * ln(abs((1 + x) / (1 -x)))+C}}$</p>\n  <p>where we have used partial fractions to integrate $\\displaystyle{\\frac{1}{1-x^2}}$.</p>\n  <p>Note carefully that in this example it is good practice to take the absolute value of the argument of $\\ln$.</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/"}]}