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Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

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Indefinite integration of basic functions.

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Integrate each of the following basic functions indefinitely.

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    \n
  1. \\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx=\\simplify{{a}*x+{b}*sin(x)+c}\\]
  2. \n
  3. \\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx=\\simplify{{c/2}*x^2+{b/a}*exp({a}*x)+c}\\]
  4. \n
  5. \\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx=\\simplify{{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c}\\]
  6. \n
  7. \\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx=\\simplify{{-c}/x+{1/(a+1)}*x^{b+1}+c}\\]
  8. \n
  9. \\[\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx=\\simplify{{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)+c}\\]
  10. \n
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\\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx\\]

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

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

", "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\\]

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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\\]

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

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It is possible to write it without a fraction.

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

", "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\\]

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

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Integrate each of the following functions with the given limits.

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Answer to question 2.

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Answer to question 1.

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Answer to question 4.

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Answer to question 5.

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Answer to question 3.

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Integrate and evaluate

\n

\\[\\int_\\var{c}^\\var{c+2}\\simplify{{a}x-sin({b}x)}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{{a}x-sin({b}x)}dx\\]

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Integrate and evaluate

\n

\\[\\int_\\var{a}^\\var{a+2}\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}dx\\]

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Integrate and evaluate

\n

\\[\\int_0^\\var{b}\\simplify{{a}*exp(x/{c+1})+{d}}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{{a}*exp(x/{c+1})+{d}}\\;dx\\]

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Integrate and evaluate

\n

\\[\\int_\\var{c}^\\var{c+1}\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "With constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "{d}/{a}*ln(x)+{a}/{d*c}*sin({d}*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": ""}]}], "answer": "{d}/{a}*ln(x)+{a}/{d*c}*sin({d}*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": ""}]}], "minValue": "{a4}", "maxValue": "{a4}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "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": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Integrate and evaluate

\n

\\[\\int_0^\\var{c}\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;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": "

First integrate indefinitely

\n

\\[\\int\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;dx\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "With constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*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": ""}]}], "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*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": ""}]}], "minValue": "{a5}", "maxValue": "{a5}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "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"}, {"name": "Basic integration 1", "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": "

Indefinite integration of basic functions.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Integrate each of the following basic functions indefinitely.

", "advice": "
    \n
  1. \\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx=\\simplify{{a}*x+{b}*sin(x)+c}\\]
  2. \n
  3. \\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx=\\simplify{{c/2}*x^2+{b/a}*exp({a}*x)+c}\\]
  4. \n
  5. \\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx=\\simplify{{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c}\\]
  6. \n
  7. \\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx=\\simplify{{-c}/x+{1/(a+1)}*x^{b+1}+c}\\]
  8. \n
  9. \\[\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx=\\simplify{{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)+c}\\]
  10. \n
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "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": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

\\[\\int\\simplify[all]{{a}+{b}*cos(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": "{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\\]

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

", "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"}, {"name": "Basic integration 2", "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": "

Integrate each of the following functions with the given limits.

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Answer to question 2.

", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "(a/2*(c+2)^2+1/b*cos(b*(c+2)))-(a/2*c^2+1/b*cos(b*c))", "description": "

Answer to question 1.

", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "(d/a*ln(c+1)+a/(d*c)*sin(d*(c+1)))-(d/a*ln(c)+a/(d*c)*sin(d*c))", "description": "

Answer to question 4.

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Answer to question 5.

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Answer to question 3.

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Integrate and evaluate

\n

\\[\\int_\\var{c}^\\var{c+2}\\simplify{{a}x-sin({b}x)}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{{a}x-sin({b}x)}dx\\]

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Integrate and evaluate

\n

\\[\\int_\\var{a}^\\var{a+2}\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}\\;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": "

First integrate indefinitely

\n

\\[\\int\\simplify{{b}/x^{c+2}+{d}*sqrt(x)}dx\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "With constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "-{b}/{c+1}*x^{-c-1}+{2*d}/3*x^(3/2)+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": ""}]}], "answer": "-{b}/{c+1}*x^{-c-1}+{2*d}/3*x^(3/2)", "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": ""}]}], "minValue": "{a2}", "maxValue": "{a2}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "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": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Integrate and evaluate

\n

\\[\\int_0^\\var{b}\\simplify{{a}*exp(x/{c+1})+{d}}\\;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": "

First integrate indefinitely

\n

\\[\\int\\simplify{{a}*exp(x/{c+1})+{d}}\\;dx\\]

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Integrate and evaluate

\n

\\[\\int_\\var{c}^\\var{c+1}\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]

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First integrate indefinitely

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\\[\\int\\simplify{{d}/({a}*x)+{a}/{c}*cos({d}*x)}\\;dx\\]

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Integrate and evaluate

\n

\\[\\int_0^\\var{c}\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;dx\\]

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First integrate indefinitely

\n

\\[\\int\\simplify{x^{a}/{b}+{c}*exp(-{b}*x)-{d}}\\;dx\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "With constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*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": ""}]}], "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*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": ""}]}], "minValue": "{a5}", "maxValue": "{a5}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "3", "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"}, {"name": "Basic integration 1", "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": "

Indefinite integration of basic functions.

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Integrate each of the following basic functions indefinitely.

", "advice": "
    \n
  1. \\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx=\\simplify{{a}*x+{b}*sin(x)+c}\\]
  2. \n
  3. \\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx=\\simplify{{c/2}*x^2+{b/a}*exp({a}*x)+c}\\]
  4. \n
  5. \\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx=\\simplify{{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c}\\]
  6. \n
  7. \\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx=\\simplify{{-c}/x+{1/(a+1)}*x^{b+1}+c}\\]
  8. \n
  9. \\[\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx=\\simplify{{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)+c}\\]
  10. \n
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\\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx\\]

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

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

", "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\\]

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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\\]

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

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It is possible to write it without a fraction.

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

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

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

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