// Numbas version: exam_results_page_options
{"name": "odd power #2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Evaluate the following definite integral correct to three decimal places:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a}^\\var{b}\\var{m}sin^\\var{n}(\\var{k}t)cos^3(\\var{k}t)dt\\)</p>", "advice": "<p>\\(\\int_\\var{a}^\\var{b}\\var{m}sin^\\var{n}(\\var{k}t)cos^3(\\var{k}t)dt\\)</p>\n<p>Let \\(u=sin(\\var{k}t)\\)&nbsp; \\(\\implies \\frac{du}{dt}=\\var{k}cos(\\var{k}t)\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\frac{1}{\\var{k}cos(\\var{k}t)}du=dt\\)</p>\n<p>\\(\\int_\\var{a}^\\var{b}\\var{m}u^\\var{n}cos^3(\\var{k}t)\\frac{1}{\\var{k}cos(\\var{k}t)}du\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{\\var{k}}u^\\var{n}cos^2(\\var{k}t)du\\)</p>\n<p style=\"padding-left: 30px;\">\\(but\\)&nbsp; \\(cos^2(\\var{k}t)=1-sin^2(\\var{k}t)=1-u^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{\\var{k}}u^\\var{n}(1-u^2)du\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{\\var{k}}u^\\var{n}-\\frac{\\var{m}}{\\var{k}}u^\\simplify{{n}+2}du\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{m}}{\\var{k}}\\frac{u^{\\simplify{{n}+1}}}{\\simplify{{n}+1}}-\\frac{\\var{m}}{\\var{k}}\\frac{u^{\\simplify{{n}+3}}}{\\simplify{{n}+3}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{m}}{\\simplify{{k}*({n}+1)}}sin^{\\simplify{{n}+1}}(\\var{k}t)-\\frac{\\var{m}}{\\simplify{{k}*({n}+3)}}sin^{\\simplify{{n}+3}}(\\var{k}t)\\Big|_\\var{a}^\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{m}}{\\simplify{{k}*({n}+1)}}\\left(sin^{\\simplify{{n}+1}}(\\simplify{{k}*{b}})-sin^{\\simplify{{n}+1}}(\\simplify{{k}*{a}})\\right)-\\frac{\\var{m}}{\\simplify{{k}*({n}+3)}}\\left(sin^{\\simplify{{n}+3}}(\\simplify{{k}*{b}})-sin^{\\simplify{{n}+3}}(\\simplify{{k}*{a}})\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\simplify{{m}/({k}*({n}+1))}*\\left(\\simplify{(sin({{k}*{b}}))}^\\simplify{{{{n}+1}}}-\\simplify{(sin({{k}*{a}}))}^\\simplify{{{{n}+1}}}\\right)-\\simplify{{m}/({k}*({n}+3))}*\\left(\\simplify{(sin({{k}*{b}}))}^\\simplify{{{{n}+3}}}-\\simplify{(sin({{k}*{a}}))}^\\simplify{{{{n}+3}}}\\right)\\)</p>", "functions": {}, "preamble": {"js": "", "css": ""}, "extensions": [], "ungrouped_variables": ["m", "a", "b", "n", "k"], "variables": {"n": {"definition": "random(2..7#1)", "name": "n", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "b": {"definition": "random(1..1.7#0.1)", "name": "b", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "m": {"definition": "random(2..9#1)", "name": "m", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "a": {"definition": "random(0..0.6#0.1)", "name": "a", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}, "k": {"definition": "random(2..10#1)", "name": "k", "description": "", "group": "Ungrouped variables", "templateType": "randrange"}}, "rulesets": {}, "name": "odd power #2", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "tags": [], "parts": [{"marks": 0, "type": "gapfill", "gaps": [{"showCorrectAnswer": true, "maxValue": "{m}*((sin({k}*{b}))^{{n}+1}-(sin({k}*{a}))^{{n}+1})/({k}*({n}+1))-{m}*((sin({k}*{b}))^{{n}+3}-(sin({k}*{a}))^{{n}+3})/({k}*({n}+3))", "type": "numberentry", "precision": "3", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "scripts": {}, "mustBeReducedPC": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": true, "marks": "2", "strictPrecision": false, "allowFractions": false, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "minValue": "{m}*((sin({k}*{b}))^{{n}+1}-(sin({k}*{a}))^{{n}+1})/({k}*({n}+1))-{m}*((sin({k}*{b}))^{{n}+3}-(sin({k}*{a}))^{{n}+3})/({k}*({n}+3))", "variableReplacements": []}], "prompt": "<p>\\(Answer =\\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}