// Numbas version: exam_results_page_options
{"name": "Part Integral 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Part Integral 1", "tags": [], "metadata": {"description": "<p>Integration by parts.</p>", "licence": "All rights reserved"}, "statement": "<p>Integrate the following expression:</p>\n<p>$f_{(x)}=\\var{a}x+x+\\var{b}x^\\var{c}+sin(\\var{d}x)$</p>", "advice": "<p>\\[ f_{(x)}=\\var{a}x+x+\\var{b}x^\\var{c}+sin(\\var{d}x) \\]</p>\n<p>Each part of the expression can be integrated separately.</p>\n<p>\\[\\int{\\var{a}x}.dx=\\var{a}\\int{x}.dx=\\var{a}\\frac{x^2}{2}\\]</p>\n<p>\\[ \\int{x.dx}=\\frac{x^2}{2}\\]</p>\n<p>\\[\\int{\\var{b}x^{\\var{c}}}.dx=\\var{b}\\int{x^{\\var{c}}}.dx=\\var{b}\\frac{x^\\var{c+1}}{\\var{c+1}}\\]</p>\n<p>For $\\int{sin(\\var{d}x)}.dx$</p>\n<p>$let u = \\var{d}x$</p>\n<p>$\\frac{du}{dx}=\\var{d}&nbsp; \\therefore dx = \\frac{du}{\\var{d}}$</p>\n<p>Sub back in</p>\n<p>\\[\\int{sin(u)}.\\frac{du}{\\var{d}}=\\frac{1}{\\var{d}}\\int{sin(u).du}=\\frac{-cos(u)}{\\var{d}}=\\frac{-cos(\\var{d}x)}{\\var{d}} \\]</p>\n<p></p>\n<p>Recombine and add the constant of integration</p>\n<p>\\[ \\var{a}\\frac{x^2}{2}+ \\frac{x^2}{2}&nbsp;+ \\var{b}\\frac{x^\\var{c+1}}{\\var{c+1}} + \\frac{-cos(\\var{d}x)}{\\var{d}} +C\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9 .. -2#1)", "description": "", "templateType": "randrange", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Use $C$ as the constant of integration.&nbsp;Ensure variables and functions are in the numerator of any fractions.</p>", "answer": "({a}*x^2)/2+x^2/2+({b}x^{c+1})/{c+1}-cos({d}x)/{d}+C", "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": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}]}], "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}