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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.

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\\[\\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.

", "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\\]

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.

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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

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

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First integrate indefinitely

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

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Integrate and evaluate

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

First integrate indefinitely

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

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Integrate and evaluate

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

First integrate indefinitely

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

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Integrate and evaluate

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

First integrate indefinitely

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

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Integrate and evaluate

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

First integrate indefinitely

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

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Consider the following integral:

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

This may be evaluated by using integration by parts.

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The product $\\simplify{{a}x*cos({b}*x)}$ must be written in the form $u\\frac{dv}{dv}$.

Determine $u,\\frac{du}{dx},\\frac{dv}{dx}$ and $v$:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$u=$	[[0]]	$\\frac{du}{dx}=$	[[1]]
$v=$	[[3]]	$\\frac{dv}{dx}=$	[[2]]

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Apply the integration by parts formula:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

$\\int{u\\frac{dv}{dv}dx}=$	$uv$	$-\\int$	$v\\frac{du}{dx}$	$dx$
$\\int{u\\frac{dv}{dv}dx}=$	[[0]]	$-\\int$	[[1]]	$dx$

Hence write down the indefinite integral:

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

As this is an indefinite integral the constant of integration $+c$ is required.

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Evaluate the definite integral:

\\[\\int_0^1\\simplify{{a}x*cos({b}*x)}\\,dx\\]

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The following integral can be evaluated by using substitution:

\\[\\int\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]

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Identify the inside function: $u=$ [[0]]

Differentiate: $\\frac{du}{dx}=$ [[1]]

Make $dx$ the subject: $dx=$ [[2]]

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Rewrite the whole integral in terms of $u$ and $du$: $\\int$ [[0]]	$du$
	[[1]]

Simplify and cancel $x$'s: $\\int$ [[2]] $du$

Integrate with respect to $u$: [[3]]

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Hence write down the indefinite integral:

\\[\\int\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]

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Replace $u$ with $\\simplify{x^({n}+1)+{b}}$ in the previous step.

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This is an indefinite integral so the constant of integration is needed.

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Evaluate the definite integral:

\\[\\int_0^\\pi\\simplify{{a}*x^{n}*sin(x^({n}+1)+{b})}\\,dx\\]

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First test

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