// Numbas version: exam_results_page_options
{"name": "Integration by parts 2 - cos and sin", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<h4>The formula for integrating by parts is</h4>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n<p>We choose $u = \\simplify[std]{({a}x+{b})}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{cos({c}*x+{d})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx} = \\simplify[std]{{a}}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{c})*sin({c}*x+{d})}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*cos({c}*x+{d})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(({a}*x+{b})/{c})*sin({c}*x+{d}) - ({a}/{c})*Int(sin({c}*x+{d}),x)} \\\\ &amp;=&amp; \\simplify[std]{(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}) + C} \\end{eqnarray} \\]</p>\n<h4></h4>", "parts": [{"unitTests": [], "customName": "", "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "{a}x+{b}", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": false, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": 1, "useCustomName": false, "answerSimplification": "all"}, {"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "cos({c}x+{d})", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": false, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": 1, "useCustomName": false, "answerSimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx $</p>\n<p>The formula for integration by parts is</p>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n<p>What is the most suitable choice for $u$ and $\\frac{dv}{dx}$?</p>\n<p>$u =\\;$[[0]]</p>\n<p>$\\frac{dv}{dx} =\\;$[[1]]</p>\n<p></p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}, {"unitTests": [], "customName": "", "marks": 0, "gaps": [{"unitTests": [], "checkingAccuracy": 0.001, "customName": "", "marks": 1, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "checkVariableNames": false, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "valuegenerators": [], "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "useCustomName": false, "answer": "{a}", "vsetRangePoints": 5}, {"unitTests": [], "checkingAccuracy": 0.001, "customName": "", "marks": 1, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "checkVariableNames": false, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "useCustomName": false, "answer": "1/{c}sin({c}x+{d})", "vsetRangePoints": 5}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Hence find $\\frac{du}{dx} =\\;$[[0]]</p>\n<p>$v =\\;$[[1]]</p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}, {"unitTests": [], "customName": "", "marks": 0, "gaps": [{"unitTests": [], "checkingAccuracy": 0.001, "customName": "", "marks": 1, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "checkVariableNames": false, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "useCustomName": false, "answer": "({a}x+{b})/{c}sin({c}x+{d})", "vsetRangePoints": 5}, {"unitTests": [], "checkingAccuracy": 0.001, "customName": "", "marks": 1, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "checkVariableNames": false, "customMarkingAlgorithm": "", "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "useCustomName": false, "answer": "-{a}/{c}^2cos({c}x+{d})", "vsetRangePoints": 5}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Hence find $uv =\\;$[[0]]</p>\n<p>$\\int v\\frac{du}{dx}\\mathrm{d}x = \\;$[[1]]$+C$</p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}, {"unitTests": [], "customName": "", "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "({a}x+{b})/{c}sin({c}x+{d})+{a}/{c}^2cos({c}x+{d})", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": false, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": "2", "useCustomName": false, "answerSimplification": "std"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Use the results from above&nbsp;to find:</p>\n<p>$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx =&nbsp;\\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx = \\;$[[0]]$+C$</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}], "variable_groups": [], "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "name": "Integration by parts 2 - cos and sin", "extensions": [], "tags": [], "metadata": {"description": "<p>Find $\\displaystyle \\int (ax+b)\\cos(cx+d)\\; dx $</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variables": {"d": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "d"}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s1"}, "c": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "c"}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "s2"}, "a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a"}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Find the following indefinite integral.</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p>Input the constant of integration as $C$.</p>", "preamble": {"css": "", "js": ""}, "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}