// Numbas version: finer_feedback_settings
{"name": "Find indefinite integrals of hyperbolic functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "name": "b", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "name": "b1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "a1", "description": ""}}, "ungrouped_variables": ["a", "b", "s2", "s1", "a1", "b1"], "name": "Find indefinite integrals of hyperbolic functions", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$f(x)=\\simplify[std]{ cosh({a}x+{b})}$</p>\n<p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n<p><em>You must include the constant of integration as $C$.</em></p>\n<p><em>Input all numbers as integers or fractions &ndash; not as decimals.</em></p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "(1 / {a}) * sinh({a} * x + {b})+C", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": [], "unitTests": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "marks": 3, "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$f(x)=\\simplify[std]{x*sinh({a1}x+{b1})}$</p>\n<p>$\\displaystyle{\\int f(x)\\;dx=\\;\\;}$[[0]]</p>\n<p><em>Include the constant of integration as $C$.</em></p>\n<p><em>Input all numbers as integers or fractions &ndash; not as decimals.</em></p>\n<p><em>Please note that if you want to enter a function of the form $xf(x)$ then enter as $x*f(x)$.</em></p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "(1 / {a1}) * x * cosh({a1} * x + {b1}) - (1 / ({a1} ^ 2)) * sinh({a1} * x + {b1})+C", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "expectedVariableNames": ["x", "c"], "unitTests": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": true, "vsetRange": [0, 1], "marks": 3, "showFeedbackIcon": true, "scripts": {}, "answerSimplification": "std", "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n\n\n<p>Integrate the following functions $f(x)$.</p>\n\n\n", "tags": ["Calculus", "calculus", "checked2015", "cosh", "hyperbolic functions", "indefinite integration", "integrating hyperbolic functions", "integration", "integration by parts", "MAS1601", "mas1601", "sinh"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find &nbsp;$\\displaystyle \\int\\cosh(ax+b)\\;dx,\\;\\;\\int x\\sinh(cx+d)\\;dx$.</p>\n<p>Advice tidied up.</p>"}, "advice": "<p>a) &nbsp;Since $\\int \\cosh(x)=\\sinh(x)+C$ it follows that:<br/>\\[\\int \\simplify[std]{cosh({a}x+{b})}\\;dx = \\frac{1}{\\var{a}}\\simplify[std]{ sinh({a}x+{b})}+C\\]</p>\n<p>b) &nbsp;We perform integration by parts:</p>\n<p>Using $\\int u dv = uv - \\int v du$ where:</p>\n<p>\\[\\begin{eqnarray*}&nbsp; &amp;u&amp;=x &nbsp; &nbsp; \\Rightarrow &nbsp; &nbsp; du = dx \\\\&nbsp; &amp;dv&amp; = \\simplify[std]{sinh({a1}x+{b1})}&nbsp; &nbsp;\\Rightarrow &nbsp; &nbsp; v= \\simplify[std]{((cosh({a1}x+{b1}))/{a1})}\\end{eqnarray*} \\]&nbsp;</p>\n<p>Hence we have:<br/>\\[\\begin{eqnarray*} \\int \\simplify[std]{x*sinh({a1}x+{b1})}\\;dx&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>", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}