// Numbas version: exam_results_page_options
{"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by substitution", "tags": ["Calculus", "calculus", "integration", "integration by substitution", "standard integrals", "substitution", "Substitution"], "metadata": {"description": "<p>Use a substitution to simplify an integral.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider the indefinite integral defined by&nbsp;<br/>\\[I=\\int\\var{b*c}x^{\\var{b-1}} \\sin(\\var{a}x^\\var{b}) dx\\]</p>\n<p>By using the substituion $u=\\var{a}x^\\var{b}$ this integral may be written as which of the following forms?</p>", "advice": "", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 1, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["\\[I=\\int\\frac{\\var{c}}{\\var{a}}\\sin(u) d u\\]", "\\[I=\\int \\var{a} \\sin(\\var{a}u) d u\\]", "None of the other options.", "\\[I=\\int \\frac{\\var{a}u}{x} \\sin(u) d u\\]"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Bissell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18193/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "John Bissell", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18193/"}]}