// Numbas version: finer_feedback_settings
{"name": "Integrate some hyperbolic functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integrate some hyperbolic functions", "tags": ["Calculus", "calculus", "cosh", "hyperbolic functions", "indefinite integration", "integral", "integrating hyperbolic functions", "integration", "integration by parts", "sinh"], "metadata": {"description": "<p>Find &nbsp;$\\displaystyle \\int\\cosh(ax+b)\\;dx,\\;\\;\\int x\\sinh(cx+d)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n    \n    \n    <p>Integrate the following functions $f(x)$.</p>\n    \n    \n    ", "advice": "<h4>a)</h4>\n<p>\\[\\int \\simplify[std]{cosh({a}x+{b})}\\;dx = \\frac{1}{\\var{a}}\\simplify[std]{ sinh({a}x+{b})}+C\\]</p>\n<h4>b)</h4>\n<p>Integrate by parts, so that</p>\n<p>\\[\\begin{eqnarray*} \\int \\simplify[std]{x*sinh({a1}x+{b1})}\\;dx&amp;=&amp;\\int \\simplify[std]{x*d((cosh({a1}x+{b1}))/{a1})}\\\\ &amp;=&amp;\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1}} \\int \\simplify[std]{cosh({a1}x+{b1})}\\;dx\\\\ &amp;=&amp;\\frac{1}{\\var{a1}}\\simplify[std]{x*cosh({a1}x+{b1})}-\\frac{1}{\\var{a1^2}}\\simplify[std]{sinh({a1}x+{b1})}+C \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "s2", "s1", "a1", "b1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n        <p>$f(x)=\\simplify[std]{ cosh({a}x+{b})}$</p>\n      <p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n      <p>You must include the constant of integration as $C$.</p>\n      <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n        ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(1 / {a}) * sinh({a} * x + {b})+C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$f(x)=\\simplify[std]{x*sinh({a1}x+{b1})}$</p>\n<p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n<p>Include the constant of integration as $C$.</p>\n<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n<p>Please note that if you want to enter a function of the form $xf(x)$ then enter as <code>x*f(x)</code>&nbsp;so that it's clear what you&nbsp;mean.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(1 / {a1}) * x * Cosh({a1} * x + {b1}) - (1 / ({a1} ^ 2)) * Sinh({a1} * x + {b1})+C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Will Roberts", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/32402/"}], "resources": []}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Will Roberts", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/32402/"}]}