// Numbas version: exam_results_page_options {"showQuestionGroupNames": false, "name": "Maria's copy of Integration practice", "percentPass": "60", "duration": 0, "navigation": {"allowregen": true, "showfrontpage": true, "browse": true, "showresultspage": "oncompletion", "onleave": {"action": "none", "message": ""}, "preventleave": true, "reverse": true}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Integration techniques", "pickQuestions": 1, "questions": [{"name": "Basic integration 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "preamble": {"js": "", "css": ""}, "variables": {"c": {"definition": "random(-5..5 except [-1,0])", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c"}, "b": {"definition": "random(2..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "d": {"definition": "random(1..15)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d"}, "a": {"definition": "random(1..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}}, "variable_groups": [], "parts": [{"prompt": "

\$\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx\$

", "checkvariablenames": false, "expectedvariablenames": [], "type": "jme", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": "2", "checkingtype": "absdiff", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x+{b}*sin(x)+c", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showpreview": true}, {"prompt": "

\$\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx\$

", "checkvariablenames": false, "expectedvariablenames": [], "type": "jme", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": "2", "checkingtype": "absdiff", "showCorrectAnswer": true, "scripts": {}, "answer": "{c}*x^2/2+{b/a}*exp({a}*x)+c", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showpreview": true}, {"prompt": "

\$\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx\$

", "checkvariablenames": false, "expectedvariablenames": [], "type": "jme", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": "2", "checkingtype": "absdiff", "showCorrectAnswer": true, "scripts": {}, "answer": "{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showpreview": true}, {"prompt": "

\$\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx\$

", "checkvariablenames": false, "expectedvariablenames": [], "type": "jme", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "stepsPenalty": "0", "marks": "2", "checkingtype": "absdiff", "steps": [{"prompt": "

Can you write $\\simplify{{c}/x^2}$ by using a negative power?

", "checkvariablenames": false, "expectedvariablenames": [], "type": "jme", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "notallowed": {"partialCredit": 0, "message": "

It is possible to write it without a fraction.

", "strings": ["/"], "showStrings": true}, "checkingtype": "absdiff", "showCorrectAnswer": true, "scripts": {}, "answer": "{c}*x^-2", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showpreview": true}], "showCorrectAnswer": true, "scripts": {}, "answer": "{-c}/x+{1/(a+1)}*x^{b+1}+c", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "vsetrangepoints": 5, "showpreview": true}, {"prompt": "

\$\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx\$

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Integrate each of the following basic functions indefinitely.

\n

Don't forget the constant of integration.

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Integrate each of the following basic functions first indefinitely and then evaluate with the given limits.

\n

", "advice": "", "rulesets": {}, "variables": {"a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-c/(a*(d+3)^a)+(2*b/3)*(d+3)^(3/2)+c/(a*d^a)-(2*b/3)*d^(3/2)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "b*(c+1)*exp((a-2)/b)-a*(a-2)-b*(c+1)*exp(-2/b)-2*a", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0])", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "a*d^2/2+cos(b*d)/b-a*(d-2)^2/2-cos(b*(d-2))/b", "description": "", "templateType": "anything"}, "a5": {"name": "a5", "group": "Ungrouped variables", "definition": "(a+1)^b/(b*(b+2))-d*(a+1)-c/a*exp(-a*(a+1))-a^{b}/(b*(b+2))+d*a+c/a*exp(-a^2)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "c/b*ln(b+a)+b/(c*(a+1))*sin(c*(b+a))-c/b*ln(b)-b/(c*(a+1))*sin(c*b)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "a1", "a2", "a3", "a4", "a5"], "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": "

Integrate indefinitely

\n

\$\\int\\simplify[all]{{a}*x-sin({b}*x)}\\,dx\$

", "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": "{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, "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, "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\$

", "minValue": "{a1}", "maxValue": "{a1}", "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{{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, "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, "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": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

You have forgotten the constant of integration, which is needed when doing indefinite integration.

", "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, "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, "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\$

", "minValue": "{a2}", "maxValue": "{a2}", "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+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 $n$?

", "minValue": "1/{b}", "maxValue": "1/{b}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "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": "

What is ${\\var{c+1}}\\div\\frac{1}{\\var{b}}$?

", "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": false, "customName": "", "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": "{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, "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, "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": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

You have forgotten the constant of integration, which is needed when doing indefinite integration.

", "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, "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, "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\$

", "minValue": "{a4}", "maxValue": "{a4}", "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]{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 $a$?

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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, "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, "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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Determine the indefinite integral $\\int y\\;dt$.

\n

(No constant required.)

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Integrate over the relevant limits. Give your answer correct to at least 3 significant figures.

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Hence calculate the mean value

\n

\$m=\\frac{1}{b-a}\\int_a^b y\\;dx\$

\n

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In this question you will calculate the mean value of the function

\n

\$y=\\var{a}+\\simplify{{b}*sin(({c}*pi*t)/2)}\$

\n

between $t=\\var{ll}$ and $t=\\var{ul}$.

\n

Hint: to type $\\pi$ enter \"pi\".

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Brackets are not needed

", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "

Expand and simplify $y^2$.

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Determine $\\int y^2\\;dx$

\n

(Constant not required.)

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Hence calculate the volume of revolution around the $x$-axis,

\n

\$V_x=\\pi\\int_a^by^2\\;dx\$

\n

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Brackets are not needed

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Now expand and simplify $xy$.

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Determine $\\int xy\\;dx$

\n

(Constant not required.)

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Hence calculate the volume of revolution around the $y$-axis,

\n

\$V_y=2\\pi\\int_a^b xy\\;dx\$

\n

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The graph of the function

\n

\$y=\\simplify{{a}*x+{b}}\$

\n

between $x=0$ 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.

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Simplify $y^2$.

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Determine $\\int y^2\\;dx$

\n

(Constant not required.)

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Hence calculate the RMS value

\n

\$\\text{RMS}=\\sqrt{\\frac{1}{b-a}\\int_a^by^2\\;dx}\$

\n

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In this question you will calculate the Root Mean Square (RMS) value of the function

\n

\$y=\\simplify{{a}*exp({b}x)}\$

\n

between $x=0$ and $x=\\var{ul}$.

\n

Hint: to type $e^x$ enter \"exp(x)\".

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What is $\\int y\\;dx$ ?

\n

(No constant required.)

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Calculate the definite integral $\\int_1^\\var{c} y\\;dx$ correct to 2 decimal places.

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Simplify $xy$.

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Hence detemine the indefinite integral $\\int xy\\;dx$

\n

(No constant required.)

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Calculate the definite integral $\\int_1^\\var{c} xy\\;dx$ correct to 2 decimal places.

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Simplify $y^2$.

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Hence detemine the indefinite integral $\\int y^2\\;dx$

\n

(No constant required.)

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Calculate the definite integral $\\int_1^\\var{c} y^2\\;dx$ correct to 2 decimal places.

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Calculate the $x$-coordinate of the centroid:

\n

\$\\bar{x}=\\frac{\\int xy\\;dx}{\\int y\\;dx}\$

\n

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Calculate the $y$-coordinate of the centroid:

\n

\$\\bar{y}=\\frac{\\frac{1}{2}\\int y^2\\;dx}{\\int y\\;dx}\$

\n

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In this question you will calculate the centroid of the region bounded by the curve

\n

\$y=\\simplify{{a}*x^{b}}\$

\n

between $x=1$ and $x=\\var{c}$.

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