Indefinite integration of basic functions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Integrate each of the following basic functions indefinitely.
", "advice": "\\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx\\]
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", "useAlternativeFeedback": false, "answer": "{a}*x+{b}*sin(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{a}*x+{b}*sin(x)+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx\\]
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", "useAlternativeFeedback": false, "answer": "{c}*x^2/2+{b/a}*exp({a}*x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{c}*x^2/2+{b/a}*exp({a}*x)+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx\\]
", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration, which is needed when doing indefinite integration.
", "useAlternativeFeedback": false, "answer": "{-(c+1)/b}*cos({b}*x)-{a}*ln(x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx\\]
", "stepsPenalty": "0", "steps": [{"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": "Can you write $\\simplify{{c}/x^2}$ by using a negative power?
", "answer": "{c}*x^-2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["/"], "showStrings": true, "partialCredit": 0, "message": "It is possible to write it without a fraction.
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", "useAlternativeFeedback": false, "answer": "{-c}/x+{1/(a+1)}*x^{b+1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{-c}/x+{1/(a+1)}*x^{b+1}+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\\[\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx\\]
", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration, which is needed when doing indefinite integration.
", "useAlternativeFeedback": false, "answer": "{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Definite integration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "Definite integation of basic functions.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Integrate each of the following basic functions first indefinitely and then evaluate with the given limits.
", "advice": "(a) and (b):
\n\\[\\int_\\var{d-2}^\\var{d}\\simplify[all]{{a}*x-sin({b}*x)}\\,dx=\\left[\\simplify{{a}*x^2/2+cos({b}*x)/{b}}\\right]_\\var{d-2}^\\var{d}=\\var{a1}\\]
\n(c) and (d):
\n\\[\\int_\\var{d}^\\var{d+3}\\simplify[all]{{c}/x^{a+1}+{b}*sqrt(x)}\\,dx=\\left[\\simplify{{-c/a}/x^{a}+{2*b/3}*x^(3/2)}\\right]_\\var{d}^\\var{d+3}=\\var{a2}\\]
\n(e) and (f):
\n\\[\\int_{-2}^\\var{a-2}\\simplify[all]{{c+1}*exp(x/{b})-{a}}\\,dx=\\left[\\simplify{{b*(c+1)}*exp(x/{b})-{a}*x}\\right]_{-2}^\\var{a-2}=\\var{a3}\\]
\n(g) and (h):
\n\\[\\int_\\var{b}^\\var{b+a}\\simplify[all]{{c}/({b}*x)+{b}/{a+1}*cos({c}*x)}\\,dx=\\left[\\simplify{{c}/{b}*ln(x)+{b}/{c*(a+1)}*sin({c}*x)}\\right]_\\var{b}^\\var{b+a}=\\var{a4}\\]
\n(i) and (j):
\n\\[\\int_\\var{a}^\\var{a+1}\\simplify[all]{x^{b-1}/{b+2}-{d}+{c}*exp({-a}*x)}\\,dx=\\left[\\simplify{x^{b}/{b*(b+2)}-{d}*x-{c/a}*exp({-a}*x)}\\right]_\\var{a}^\\var{a+1}=\\var{a5}\\]
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\n\\[\\int\\simplify[all]{{a}*x-sin({b}*x)}\\,dx\\]
", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration (condoned on this occasion).
", "useAlternativeFeedback": false, "answer": "{a}*x^2/2+cos({b}*x)/{b}", "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}*x^2/2+cos({b}*x)/{b}+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": ""}]}, {"type": "numberentry", "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": "Hence evaluate
\n\\[\\int_\\var{d-2}^\\var{d}\\simplify[all]{{a}*x-sin({b}*x)}\\,dx\\]
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\n\\[\\int\\simplify{{c}/x^{a+1}+{b}*sqrt(x)}\\,dx\\]
", "stepsPenalty": 0, "steps": [{"type": "jme", "useCustomName": true, "customName": "Hint 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Write $\\simplify{{c}/x^{a+1}}$ using a negative power.
", "answer": "{c}*x^{-a-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["/"], "showStrings": false, "partialCredit": 0, "message": "Can you write it without using a fraction?"}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Hint 2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Write $\\simplify{{b}*sqrt(x)}$ without using a square root sign.
", "answer": "{b}*x^{0.5}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["sqrt"], "showStrings": false, "partialCredit": 0, "message": "Can you write it without using a square root sign?"}, "valuegenerators": [{"name": "x", "value": ""}]}], "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration (condoned on this occasion).
", "useAlternativeFeedback": false, "answer": "{-c/a}/x^{a}+{2*b/3}*x^(3/2)", "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": "{-c/a}/x^{a}+{2*b/3}*x^(3/2)+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": ""}]}, {"type": "numberentry", "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": "Hence evaluate
\n\\[\\int_\\var{d}^\\var{d+3}\\simplify[all]{{c}/x^{a+1}+{b}*sqrt(x)}\\,dx\\]
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\n\\[\\int\\simplify[all]{{c+1}*exp(x/{b})-{a}}\\,dx\\]
", "stepsPenalty": "0", "steps": [{"type": "numberentry", "useCustomName": true, "customName": "Hint 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "If $\\simplify{exp(x/{b})}$ is written in form $e^{nx}$ what is the value of $n$?
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", "minValue": "{c+1}*{b}", "maxValue": "{c+1}*{b}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration (condoned on this occasion).
", "useAlternativeFeedback": false, "answer": "{b*(c+1)}*exp(x/{b})-{a}*x", "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": "{b*(c+1)}*exp(x/{b})-{a}*x+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": ""}]}, {"type": "numberentry", "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": "Hence evaluate
\n\\[\\int_{-2}^\\var{a-2}\\simplify[all]{{c+1}*exp(x/{b})-{a}}\\,dx\\]
", "minValue": "{a3}", "maxValue": "{a3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "Integrate indefinitely
\n\\[\\int\\simplify[all]{{c}/({b}*x)+{b}/{a+1}*cos({c}*x)}\\,dx\\]
", "stepsPenalty": "", "steps": [{"type": "information", "useCustomName": true, "customName": "Hint", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\simplify[all]{{c}/({b}*x)}$ can be rewritten as $\\simplify[all]{{c}/{b}*x^{-1}}$.
"}], "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration (condoned on this occasion).
", "useAlternativeFeedback": false, "answer": "{c}/{b}*ln(x)+{b}/{c*(a+1)}*sin({c}*x)", "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": "{c}/{b}*ln(x)+{b}/{c*(a+1)}*sin({c}*x)+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": ""}]}, {"type": "numberentry", "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": "Hence evaluate
\n\\[\\int_\\var{b}^\\var{b+a}\\simplify[all]{{c}/({b}*x)+{b}/{a+1}*cos({c}*x)}\\,dx\\]
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\n\\[\\int\\simplify[all]{x^{b-1}/{b+2}-{d}+{c}*exp({-a}*x)}\\,dx\\]
", "stepsPenalty": "", "steps": [{"type": "numberentry", "useCustomName": true, "customName": "Hint", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "If $\\simplify[all]{x^{b-1}/{b+2}}$ is written in the form $\\simplify{a*x^{b-1}}$, what is the value of $a$?
", "minValue": "{1/(b+2)}", "maxValue": "{1/(b+2)}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You have forgotten the constant of integration, which is needed when doing indefinite integration.
", "useAlternativeFeedback": false, "answer": "x^{b}/{b*(b+2)}-{d}*x-{c/a}*exp({-a}*x)", "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": "x^{b}/{b*(b+2)}-{d}*x-{c/a}*exp({-a}*x)+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": ""}]}, {"type": "numberentry", "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": "Hence evaluate
\n\\[\\int_\\var{a}^\\var{a+1}\\simplify[all]{x^{b-1}/{b+2}-{d}+{c}*exp({-a}*x)}\\,dx\\]
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\n\\[y=\\var{a}+\\simplify{{b}*sin(({c}*pi*t)/2)}\\]
\nbetween $t=\\var{ll}$ and $t=\\var{ul}$.
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\nHint: to type $\\pi$ enter \"pi\".
", "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "Correct. You included the constant of integration which is not needed in this question.
", "useAlternativeFeedback": false, "answer": "{a}*t-{2*b}/{c*pi}*cos(({c}*pi)/2*t)+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": "t", "value": ""}]}], "answer": "{a}*t-{2*b}/{c*pi}*cos(({c}*pi)/2*t)", "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": "t", "value": ""}]}, {"type": "numberentry", "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": "Integrate $y$ over the relevant limits.
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\n\\[m=\\frac{1}{b-a}\\int_a^b y\\;dx\\]
"}], "minValue": "{m}", "maxValue": "{m}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "2", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Volumes of revolution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "The graph of the function
\n\\[y=\\simplify{{a}*x+{b}}\\]
\nbetween $x=0$ and $x=\\var{ul}$ is revolved firstly around the $x$-axis and then around the $y$-axis. In this question you will calculate the volume of each shape.
", "advice": "Start by expanding $y^2$ and simplifying:
\n\\[y^2=\\simplify[!cancelTerms]{({a}*x+{b})({a}*x+{b})}=\\simplify[!cancelTerms]{{a^2}*x^2+{a*b}x+{a*b}x+{b^2}}=\\simplify{{a^2}*x^2+{2*a*b}x+{b^2}}\\]
\nIntegrate:
\n\\[\\int y^2\\,dx=\\simplify[fractionNumbers]{{a^2/3}*x^3+{a*b}x^2+{b^2}x+c}\\]
\n(Note: the $+c$ may be omitted in this question.) Evaluate at upper limit:
\n\\[\\var[fractionNumbers]{a^2/3}\\times\\var{ul}^3-\\var{-a*b}\\times\\var{ul}^2+\\var{b^2}\\times\\var{ul}=\\var{siground(intx,5)}\\]
\nEvaluate at lower limit:
\n\\[\\var[fractionNumbers]{a^2/3}\\times0^3-\\var{-a*b}\\times0^2+\\var{b^2}\\times0=0\\]
\nTherefore
\n\\[V_x=\\pi\\int_0^\\var{ul}y^2\\,dx=\\pi(\\var{siground(intx,5)}-0)=\\var{siground(vx,5)}\\]
\nRounding to 3 significant figures gives $V_x=\\var{siground(vx,3)}$
\nStart by expanding $xy$ and simplifying:
\n\\[xy=\\simplify[!cancelTerms]{x({a}*x+{b})}=\\simplify{{a}*x^2+{b}x}\\]
\nIntegrate:
\n\\[\\int xy\\,dx=\\simplify[all,fractionNumbers]{{a/3}*x^3+{b/2}x^2+c}\\]
\n(Note: the $+c$ may be omitted in this question.) Evaluate at upper limit:
\n\\[\\var[fractionNumbers]{a/3}\\times\\var{ul}^3+\\var[fractionNumbers]{b/2}\\times\\var{ul}^2=\\var{siground(inty,5)}\\]
\nEvaluate at lower limit:
\n\\[\\var[fractionNumbers]{a/3}\\times0^3+\\var[fractionNumbers]{b/2}\\times0^2=0\\]
\nTherefore
\n\\[V_y=2\\pi\\int_0^\\var{ul}xy\\,dx=2\\pi(\\var{siground(inty,5)}-0)=\\var{siground(vy,5)}\\]
\nRounding to 3 significant figures gives $V_y=\\var{siground(vy,3)}$
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\n\\[V_x=\\pi\\int_a^by^2\\;dx\\]
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", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "With constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "Your answer is correct. You included the constant of integration, which is not needed in this question.
", "useAlternativeFeedback": false, "answer": "{a/3}*x^3+{b/2}*x^2+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": ""}]}], "answer": "{a}/3*x^3+{b}/2*x^2", "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": ""}]}, {"type": "numberentry", "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": "Hence calculate the volume of revolution around the $y$-axis.
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\n\\[V_y=2\\pi\\int_a^b xy\\;dx\\]
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\n\\[y=\\simplify{{a}*exp({b}x)}\\]
\nbetween $x=0$ and $x=\\var{ul}$.
\nHint: to type $e^x$ enter \"exp(x)\".
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\n\\[\\text{RMS}=\\sqrt{\\frac{1}{b-a}\\int_a^by^2\\;dx}\\]
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\n\\[y=\\simplify{{a}*x^{b}}\\]
\nbetween $x=1$ and $x=\\var{c}$.
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\n\\[\\int_1^\\var{c} y\\;dx\\]
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\n\\[\\int_1^\\var{c} xy\\;dx\\]
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\n\\[\\int_1^\\var{c} y^2\\;dx\\]
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\n\\[\\bar{x}=\\frac{\\int xy\\;dx}{\\int y\\;dx}\\]
"}], "minValue": "{xbar}", "maxValue": "{xbar}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "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": "Calculate the $y$-coordinate of the centroid.
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\n\\[\\bar{y}=\\frac{\\frac{1}{2}\\int y^2\\;dx}{\\int y\\;dx}\\]
"}], "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "Forgot to halve", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "You may have omitted to multiply by $\\frac{1}{2}$
", "useAlternativeFeedback": false, "minValue": "{ybar*2}", "maxValue": "{ybar*2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{ybar}", "maxValue": "{ybar}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}], "showQuestionGroupNames": false, "metadata": {"description": "This quiz is designed to help you practise your integration techniques.
\nImprovements needed:
\n