// Numbas version: finer_feedback_settings
{"name": "Hyperbolic Functions 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "s2", "s1", "a1", "b1"], "name": "Hyperbolic Functions 3", "tags": ["Calculus", "calculus", "cosh", "hyperbolic functions", "indefinite integration", "integral", "integrating hyperbolic functions", "integration", "integration by parts", "sinh"], "preamble": {"css": "", "js": ""}, "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"]}, "parts": [{"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        ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 / {a}) * sinh({a} * x + {b})+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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>", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(1 / {a1}) * x * Cosh({a1} * x + {b1}) - (1 / ({a1} ^ 2)) * Sinh({a1} * x + {b1})+C", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}], "statement": "\n    \n    \n    <p>Integrate the following functions $f(x)$.</p>\n    \n    \n    ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "a1": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "b1": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>29/06/2012:</strong></p>\n        \t\t    <p><strong><br /></strong>Added and edited tags.</p>\n        \t\t    <p>Added forbidden strings, the decimal point, to ensure fraction or integer input for numbers.</p>\n        \t\t    <p>Put in $C$ as integration constant in both answers.</p>\n        \t\t    <p><strong>19/07/2012:</strong></p>\n        \t\t    <p>Added instruction in second question so that an input of the form $x\\cosh$ is recognized.</p>\n        \t\t    <p>Added description.</p>\n        \t\t    <p>Checked calculation.</p>\n        \t\t    <p><strong>23/07/2012:</strong></p>\n        \t\t    <p></p>\n        \t\t    <p>Added tags.</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t    <p>&nbsp;</p>\n        \t\t", "description": "<p>Find &nbsp;$\\displaystyle \\int\\cosh(ax+b)\\;dx,\\;\\;\\int x\\sinh(cx+d)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}