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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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\\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx=\\simplify{{a}*x+{b}*sin(x)+c}\\]
\\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx=\\simplify{{c/2}*x^2+{b/a}*exp({a}*x)+c}\\]
\\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx=\\simplify{{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c}\\]
\\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx=\\simplify{{-c}/x+{1/(a+1)}*x^{b+1}+c}\\]
\\[\\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}\\]

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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\\]

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

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

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

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\\[\\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\\]

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": "", "pickingStrategy": "all-ordered"}], "variable_groups": [], "showQuestionGroupNames": false, "name": "prova", "metadata": {"licence": null}, "type": "question", "contributors": [{"name": "riccardo belle'", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1797/"}], "extensions": [], "custom_part_types": [], "resources": []}