// Numbas version: finer_feedback_settings
{"name": "Integration (Negative and Fractional Powers) 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration (Negative and Fractional Powers) 1", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Using the Table of Integrals/Antiderivatives, calculate the&nbsp;integral of $y=\\simplify[all,fractionNumbers]{{a}x^{b/c}}$.</p>\n<p></p>", "advice": "<p>From the Table of Integrals&nbsp;we see that a function of the form \\[ f(x)=x^n \\] has the integral \\[ \\int x^n dx&nbsp; =&nbsp; \\frac{x^{n+1}}{n+1}+ c,\\]</p>\n<p>and \\[\\int kf(x) dx = k \\int f(x) dx.\\]</p>\n<p>So, for the function</p>\n<p>\\[ y=\\simplify[fractionNumbers]{{a}x^{b/c}}, \\]</p>\n<p>the&nbsp;integral &nbsp;is</p>\n<p>\\[ \\begin{split}\\simplify[fractionNumbers]{int({a}x^{b/c},x)} = \\simplify[all,fractionNumbers]{{a}int(x^{b/c},x)}&amp;\\,=\\simplify[fractionNumbers,alwaysTimes]{{a}{c/(b+c)}} \\simplify[all,fractionNumbers,!collectNumbers,!simplifyFractions,!expandBrackets]{x^({b/c+1})} + c,\\\\ \\\\&amp;\\,= \\simplify[all,fractionNumbers]{{a/(b/c+1)}x^{b/c+1}}+c.\\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b<c and b/c<>-1 ", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "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, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You forgot the constant of integration</p>", "useAlternativeFeedback": false, "answer": "{a*c}x^{(b+c)/c}/{b+c}", "answerSimplification": "fractionNumbers", "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": "x", "value": ""}]}], "answer": "{(a*c)/(b+c)}x^{(b+c)/c}+c", "answerSimplification": "all,fractionNumbers,!noLeadingMinus", "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/"}], "resources": []}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}