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{"name": "Integration: By Parts 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration: By Parts 5", "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\ln(ax)$ using integration by parts.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the integral \\[ \\simplify{int(ln({a}x) ,x)}\\]</p>", "advice": "<p>To calculate the integral</p>\n<p>\\[ \\simplify{int(ln({a}x),x)},\\] despite there only being a single function of $x$, we can make use of integration by parts. This is because there is no 'standard' integral for $\\simplify{ln({a}x)}$.</p>\n<p>However, if we think of&nbsp;\\[ \\simplify{int(ln({a}x),x)} = \\int 1 \\times \\ln(\\var{a}x)\\, \\text{d}x ,\\]</p>\n<p>we can set \\[ u=\\simplify{ln({a}x)}, \\quad \\frac{dv}{dx}=1,\\]</p>\n<p>so we only have to&nbsp;<em>differentiate</em> the $\\simplify{ln({a}x)}$ term.</p>\n<p>Now, calculating the&nbsp;$\\tfrac{du}{dx}$ and $v(x)$ terms:</p>\n<p>\\[ \\begin{split} u(x) =&nbsp;\\simplify{ln({a}x)} \\quad &amp;\\implies \\frac{du}{dx} = \\simplify{1/x}; \\\\ \\frac{dv}{dx} =&nbsp;1 &amp;\\implies v(x) = x. \\end{split} \\]</p>\n<p>Plugging these 4 terms into the integration by parts formula we can succesfully find a solution:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int(ln({a}x),x)} &amp;\\,= \\simplify{x ln({a}x)}&nbsp;- \\simplify[alwaysTimes]{int((1/x)x,x)}, \\\\ &amp;\\,=\\simplify{x ln({a}x)-int(1,x)}, \\\\ &amp;\\,= \\simplify{x ln({a}x) - x}+c, \\\\ &amp;\\,=\\simplify{x (ln({a}x)-1)}+c.\\end{split} \\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "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>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x*(ln({a}x)-1)+c", "answerSimplification": "all", "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": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}