// Numbas version: finer_feedback_settings
{"name": "Integrating Hyperbolics 3&4 combined", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integrating Hyperbolics 3&4 combined", "tags": [], "metadata": {"description": "<p>Rearrange and make a substitution to integrate a quotient to a function in terms of arsinh.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Rearrange and make an appropriate substitutions to evaluate the following integrals.</p>\n<p>Don't forget to include '$+C$'!</p>", "advice": "<p>We will be using the identity</p>\n<p style=\"text-align: center;\">$\\int \\simplify{1/sqrt(x^2 + a^2)} \\, dx = \\simplify{arsinh(x/a)} + c$</p>\n<p style=\"text-align: left;\">Therefore, we have</p>\n<p style=\"text-align: center;\">$\\begin{aligned}<br/>y &amp;= \\int \\simplify{1/sqrt(x^2 + {2a}x + {a^2} + {b^2})} \\,dx \\\\<br/>&amp;= \\int \\simplify{1/sqrt((x + {a})^2 + {b^2})} \\, dx \\\\<br/>\\end{aligned}$</p>\n<p style=\"text-align: left;\">We can then set $u = x + \\var{a}$ and $a = \\var{b}$ and have</p>\n<p style=\"text-align: center;\">$\\begin{aligned}<br/>y &amp;= \\int \\simplify{1/sqrt(x^2 + {2a}x + {a^2} + {b^2})} \\, dx \\\\<br/>&amp;= \\int \\simplify{1/sqrt(u^2 + a^2)} \\, dx \\\\<br/>&amp;= \\simplify{arsinh(u/a) + c} \\\\<br/>&amp;= \\simplify{arsinh((x+{a})/{b}) + c}&nbsp;<br/>\\end{aligned}$</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..6)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..5)^2", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "c * random(2..5)^2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d"], "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;\">$<br/>\\int \\simplify{1/sqrt(x^2 + {2a}x + {a^2} + {b^2})} \\, dx = $ [[0]]</p>\n<p></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": "arsinh((x + {A})/{B}) + c", "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": "arsinh", "value": ""}, {"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 style=\"text-align: left;\">$\\int \\simplify{sqrt({a}x^2 + {b})} \\, dx =$[[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": "{sqrt(d)/2}(arsinh(x/{sqrt(d/c)}) + x/{sqrt(d/c)}sqrt( (1 + (x/{sqrt(d/c)})^2)))", "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": "arsinh", "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/"}]}