// Numbas version: finer_feedback_settings
{"metadata": {"notes": "", "description": "<p>Find the integral of an improper fraction.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "navigation": {"showfrontpage": true, "reverse": true, "onleave": {"action": "none", "message": ""}, "browse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true}, "name": "Maria's copy of Integration of improper fractions", "showQuestionGroupNames": false, "pickQuestions": 0, "duration": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "percentPass": 0, "question_groups": [{"name": "", "questions": [{"name": "Indefinite integral of improper polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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>"}, {"name": "Indefinite integral of improper polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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": "<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}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>", "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": "<p><strong>29/06/2012:</strong></p>\n        <p><strong><br /></strong>Added tags. Tidied up display of prompt.</p>\n        <p><strong>19/07/2012:</strong></p>\n        <p>Added description.</p>\n        <p>Checked calculation.</p>\n        <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        <p><strong>23/07/2012:</strong></p>\n        <p></p>\n        <p>Solution always requires arctan(x) and not arcsin(x) or arccos(x). Is this on purpose?</p>\n        <p>&nbsp;</p>\n        <p>Question appears to be working correctly.</p>\n        <p>&nbsp;</p>", "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": "<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>"}, {"name": "Integral of improper polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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>"}, {"name": "Integral of improper polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "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}], "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    "}], "pickQuestions": 0, "pickingStrategy": "all-ordered"}], "questions": [], "type": "exam", "feedback": {"showtotalmark": true, "showactualmark": true, "advicethreshold": 0, "showanswerstate": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "shuffleQuestions": false, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": []}