// Numbas version: exam_results_page_options
{"name": "Jo-Ann's copy of Integration by parts - logarithm", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "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 $u = \\simplify[std]{ln{c}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{{a}x}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx} = \\simplify[std]{1/x}$ and $\\displaystyle&nbsp;v = \\simplify[std]{({a}x^2/2)}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x)*ln({c}*x)} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(({a}*x^2)/2)*ln({c}*x) - Int(({a}*x/2),x)} \\\\ &amp;=&amp; \\simplify[std]{(({a}x^2)/2)*ln({c}*x) -({a}x^2/4) + C} \\end{eqnarray} \\]</p>\n<h4></h4>", "preamble": {"css": "", "js": ""}, "variables": {"b": {"description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables", "name": "b", "templateType": "anything"}, "s1": {"description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s1", "templateType": "anything"}, "d": {"description": "", "definition": "s2*random(1..9)", "group": "Ungrouped variables", "name": "d", "templateType": "anything"}, "a": {"description": "", "definition": "random(2..5)", "group": "Ungrouped variables", "name": "a", "templateType": "anything"}, "s2": {"description": "", "definition": "random(1,-1)", "group": "Ungrouped variables", "name": "s2", "templateType": "anything"}, "c": {"description": "", "definition": "random(2..5)", "group": "Ungrouped variables", "name": "c", "templateType": "anything"}}, "extensions": [], "parts": [{"scripts": {}, "unitTests": [], "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "ln({c}x)", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}, {"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "{a}x", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "all", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>$I=\\displaystyle \\int \\simplify[std]{({a}x)*ln({c}x)} 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>", "showCorrectAnswer": true}, {"scripts": {}, "unitTests": [], "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "1/x", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}, {"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "{a}x^2/2", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Hence find:</p>\n<p>$\\frac{du}{dx} =\\;$[[0]]</p>\n<p>$v =\\;$[[1]]</p>", "showCorrectAnswer": true}, {"scripts": {}, "unitTests": [], "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "{a}x^2/2ln({c}x)", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}, {"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": 1, "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "{a}x^2/4", "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Hence find:</p>\n<p>$uv =\\;$[[0]]</p>\n<p>$\\int v\\frac{du}{dx}\\mathrm{d}x = \\;$[[1]]$+C$</p>", "showCorrectAnswer": true}, {"scripts": {}, "unitTests": [], "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "expectedVariableNames": [], "checkVariableNames": false, "scripts": {}, "showPreview": true, "vsetRangePoints": 5, "unitTests": [], "marks": "2", "failureRate": 1, "vsetRange": [0, 1], "variableReplacementStrategy": "originalfirst", "answer": "{a}x^2/2ln({c}x)-{a}x^2/4", "notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "type": "jme", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "showFeedbackIcon": true, "checkingType": "absdiff", "showCorrectAnswer": true}], "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Use the results from above&nbsp;to find:</p>\n<p>$I=\\displaystyle \\int \\simplify[std]{({a}x)*ln({c}x)} dx =&nbsp; uv - \\int v \\frac{du}{dx} dx = \\;$[[0]]$+C$</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>", "showCorrectAnswer": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Find the following indefinite integral.</p>\n<p>This question is scaffolded - i.e. it takes you through answering the question step by step.</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p>Don't forget&nbsp;$C$!</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": [], "name": "Jo-Ann's copy of Integration by parts - logarithm", "metadata": {"description": "<p>Find $\\displaystyle \\int (ax)\\ln(cx)\\; dx $</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1"], "type": "question", "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}, {"name": "Jo-Ann Lyons", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2630/"}]}