// Numbas version: exam_results_page_options
{"name": "Integration by parts 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by parts 2", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "advice": "<p>Example,&nbsp;</p>\n<p>$\\displaystyle \\int 2xe^x\\,dx$<br/>$u = 2x \\rightarrow \\frac{du}{dx}=2$<br/>$\\frac{dv}{dx}=e^x \\rightarrow v = \\int e^x.dx = e^x$<br/>$\\int 2xe^x.dx = (2x)\\left(e^x\\right) - \\int \\left(e^x \\right) \\left(2\\right).dx$<br/>$=2xe^x - 2\\int e^x.dx$<br/>$=2xe^x - 2e^x + C$<br/>$=2e^x(x-1) + C$</p>\n<p>The tabular method could also be used. In this approach, one function is differentiated until it reaches zero. The other is integrated the corresponding number of times. An alternative sign approach is then used to multiple down the table and form the answer. Please see the notes from AMR129 for an example of this.&nbsp;</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve the following integral using the 'by parts' method:</p>\n<p>$\\int xe^{3x}\\, dx$</p>\n<p>Enter your answer in the same way you would in MATLAB.</p>\n<p>Fractions can be entered using the \\ symbol.</p>\n<p>Variables must be multiplied using the * symbol, for example $xy$ should be entered as x*y</p>\n<p>Functions must include brackets, for example $\\sin{x}$ should be entered as sin(x)</p>\n<p>Powers can be entered using the ^ symbol, for example $e^x$ can be entered as e^(x)</p>\n<p>Don't forget to include the \"+ C\" as it is an indefinite integral.</p>", "answer": "x*e^(3x)/3 - e^(3x)/9+C", "answerSimplification": "basic", "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": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}]}]}], "contributors": [{"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}]}