// Numbas version: finer_feedback_settings
{"name": "Integrating Hyperbolics 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integrating Hyperbolics 2", "tags": [], "metadata": {"description": "<p>Use defintions of the hyperbolic functions to integrate by considering exponentials.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Use the algebraic definitions of the hyperbolic identities to evaluate the following integral.</p>\n<p>Don't forget to include '$+C$'!</p>", "advice": "<p>We will make use of the definition of $\\cosh(x)$, that is</p>\n<p style=\"text-align: left;\">$$<br/>\\cosh(ax) = \\frac{e^{ax} + e^{-ax}}{2}.<br/>$$</p>\n<p style=\"text-align: left;\">Therefore, we have</p>\n<p style=\"text-align: left;\">$$<br/>\\begin{aligned}<br/>y &amp;= \\int \\simplify{{A}e^({B}x) cosh({B}x)} \\, dx \\\\<br/>&amp;= \\int \\simplify{{A}e^({B}x)(e^({B}x) + e^(-{B}x))/2} \\, dx \\\\<br/>&amp;= \\simplify[all]{{A}/{2}} \\int \\simplify{e^({2B}x) + 1} \\, dx \\\\<br/>&amp;= \\simplify[all]{{A}/{4B}e^({2B}x) + {A}x/2} + C.<br/>\\end{aligned}<br/>$$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "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": "<p style=\"text-align: left;\">$$y = \\int \\simplify{{A}e^({B}x) cosh({B}x)} \\, dx$$</p>\n<p style=\"text-align: left;\">$y = $ [[0]]</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": "{A}/{4B}e^({2B}x) + {A}x/2 + c", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}, {"name": "Will Roberts", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/32402/"}], "resources": []}]}], "contributors": [{"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}, {"name": "Will Roberts", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/32402/"}]}