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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

\\[\\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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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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The following integral can be evaluated by using substitution:

\\[\\int\\simplify{{a}*x^{n}/sqrt(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]]	$\\times$	$du$
	[[4]]		[[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}/sqrt(x^({n}+1)+{b})}\\,dx\\]

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

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Don't forget the constant of integration as this is an indefinite integral.

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Evaluate the definite integral:

\\[\\int_0^{10}\\simplify{{a}*x^{n}/sqrt(x^({n}+1)+{b})}\\,dx\\]

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The following integral can be evaluated by using substitution:

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

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

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

Factorise this: $\\frac{du}{dx}=$ [[2]] $($[[3]]$)$

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Rewrite the whole integral in terms of $u$ and $du$: $\\int$	[[0]]	$du$
		[[1]] $($[[2]]$)$

Simplify and cancel $\\simplify{x^({m}-1)+{b}}$: $\\int$ [[3]] $du$

Integrate with respect to $u$: [[4]]

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Hence write down the indefinite integral:

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

Replace $u$ with $\\simplify{x^{m}+{m}*{b}*x}$ in the previous step.

You forgot the constant of integration which is needed as this is an indefinite integral.

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Evaluate the definite integral:

\\[\\int_0^1\\simplify{{a}(x^({m}-1)+{b})(x^{m}+{m}*{b}*x)^{n}}\\,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.

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\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

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

", "useAlternativeFeedback": false, "answer": "{a/b}*x*sin({b}*x)+{a/b^2}*cos({b}*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/b}*x*sin({b}*x)+{a/b^2}*cos({b}*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": "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": "

Evaluate the definite integral:

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

", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "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"}, {"name": "Integration by parts 2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]], "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 following integral may be evaluated by using integration by parts.

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

Note: $e^{\\simplify{{b}*x}}$ can be entered by typing exp($\\simplify{{b}*x}$).

The product $\\simplify{{a}x*exp({b}*x)}$ 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*exp({b}*x)}\\,dx\\]

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

", "useAlternativeFeedback": false, "answer": "{a}/{b}*x*exp({b}*x)-{a}/{b}^2*exp({b}*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}/{b}*x*exp({b}*x)-{a}/{b}^2*exp({b}*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": "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": "

Evaluate the definite integral:

\\[\\int_{-1}^1\\simplify{{a}x*exp({b}*x)}\\,dx\\]

", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "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"}, {"name": "Integration by parts 3", "extensions": [], "custom_part_types": [], "resources": [["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]], "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 following integral may be evaluated by using integration by parts.

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

Notes:

powers can be entered using ^
to divide, use /

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The product $\\simplify{{a}x^{n}*ln({b}*x)}$ 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^{n}*ln({b}*x)}\\,dx\\]

", "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

As this is an indefinite integral, $+c$ is required.

", "useAlternativeFeedback": false, "answer": "{a}*x^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}x)-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": "{a}*x^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}x)-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": "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": "

Evaluate the definite integral:

\\[\\int_1^2\\simplify{{a}x^{n}*ln({b}*x)}\\,dx\\]

", "minValue": "ans", "maxValue": "ans", "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"}]}, {"pickQuestions": 1, "name": "Partial fractions", "pickingStrategy": "all-ordered", "questions": [{"name": "Integrate with partial fractions", "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 following integral can be integrated using partial fractions:

\\[\\int\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}\\,dx\\]

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Factorise the denominator: $\\simplify[all]{x^2+{c+d}*x+{c*d}}=$ [[0]]

You should factorise and will not need to use powers.

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Select the correct form of the partial fraction:

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\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d+1}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c-1}}}+\\frac{B}{\\simplify{x+{d}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x-{d}}}\\]

"], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Cross-multiply:

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

$A$	$+$	$B$	$=$	[[0]]
$\\simplify{x+{c}}$		$\\simplify{x+{d}}$		$\\simplify{(x+{c})*(x+{d})}$

Equate the numerators of the original fraction $\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}$ and your fraction from step c):

[[0]] $=$ [[1]]

Substitute $x=\\var{-c}$ into the numerators: [[0]] = [[1]]

Therefore $A=$ [[2]]

Substitute $x=\\var{-d}$ into the numerators: [[0]] = [[1]]

Therefore $B=$ [[2]]

Therefore

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

$\\simplify[all]{{a+b}x+{ad+b*c}}$	$=$	[[0]]	$+$	[[1]]
$\\simplify[all]{x^2+{c+d}x+{cd}}$		$\\simplify{x+{c}}$		$\\simplify{x+{d}}$

", "gaps": [{"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, "minValue": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": 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, "minValue": "b", "maxValue": "b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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": "

Hence calculate the following indefinite integral:

\\[\\int\\frac{\\simplify[all]{{a+b}*x+{a*d+b*c}}}{\\simplify[all]{x^2+{c+d}*x+{c*d}}}\\,dx\\]

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

Don't forget the constant of integration as this is an indefinite integral.

", "useAlternativeFeedback": false, "answer": "{a}*ln(x+{c})+{b}*ln(x+{d})", "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}*ln(x+{c})+{b}*ln(x+{d})+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": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Integrate with partial fractions 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": "

Consider the following integral:

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

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(0..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5 except a)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "c+random(3,4,5)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "{n1}*(ln(10+{c})-ln(1+{c}))+{n2}*(ln(10+{d})-ln(1+{d}))", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "(b-a*c)/(d-c)", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "(b-a*d)/(c-d)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "15"}, "ungrouped_variables": ["a", "b", "c", "d", "ans", "n1", "n2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Select the correct form of the partial fraction:

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x+{d+1}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c-1}}}+\\frac{B}{\\simplify{x+{d}}}\\]

", "

\\[\\frac{A}{\\simplify{x+{c}}}+\\frac{B}{\\simplify{x-{d}}}\\]

Therefore, by cross-multiplication:

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

$\\simplify[all]{{a}*x+{b}}$	$=$	$A$	$+$	$B$	$=$	[[0]]
$\\simplify{(x+{c})*(x+{d})}$		$\\simplify{x+{c}}$		$\\simplify{x+{d}}$		$\\simplify{(x+{c})*(x+{d})}$

Substitute $x=\\var{-c}$ into the numerators above: [[0]] = [[1]]

Therefore $A=$ [[2]]

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b-a*c", "maxValue": "b-a*c", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({d}-{c})*A", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "a", "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, "minValue": "n1", "maxValue": "n1", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Substitute $x=\\var{-d}$ into the numerators: [[0]] = [[1]]

Therefore $B=$ [[2]]

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Hence calculate the following indefinite integral:

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

Don't forget the constant of integration as this is an indefinite integral.

", "useAlternativeFeedback": false, "answer": "{n1}*ln(x+{c})+{n2}*ln(x+{d})", "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": "{n1}*ln(x+{c})+{n2}*ln(x+{d})+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": "

Evaluate the definite integral:

\\[\\int_1^{10}\\frac{\\simplify[all]{{a}*x+{b}}}{\\simplify{(x+{c})*(x+{d})}}\\,dx\\]

", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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"}]}, {"pickQuestions": 1, "name": "Trig product", "pickingStrategy": "all-ordered", "questions": [{"name": "Integrate trig product", "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 following integral involves a trigonometric product:

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

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "2*random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(c+1..8)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "-{a}/(2({b}+{c}))*cos(({b}+{c})*pi/2)-{a}/(2({b}-{c}))*cos(({b}-{c})*pi/2)+{a}/(2({b}+{c}))+{a}/(2({b}-{c}))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Select the relevant trigonometric product identity from the list below:

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Use the identity to rewrite $\\simplify{{a}*sin({b}x)*cos({c}x)}$:

", "alternatives": [{"type": "jme", "useCustomName": true, "customName": "Did not multiply", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "

You have not multiplied by $\\var{a}$.

", "useAlternativeFeedback": false, "answer": "1/2*(sin(({b}+{c})*x)+sin(({b}-{c})*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}/2*(sin(({b}+{c})*x)+sin(({b}-{c})*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": ""}]}, {"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": "

Hence write down the indefinite integral:

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

Don't forget the constant of integration as this is an indefinite integral.

", "useAlternativeFeedback": false, "answer": "-{a}/(2({b}+{c}))*cos(({b}+{c})*x)-{a}/(2({b}-{c}))*cos(({b}-{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": "-{a}/(2({b}+{c}))*cos(({b}+{c})*x)-{a}/(2({b}-{c}))*cos(({b}-{c})*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": "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 the definite integral:

\\[\\int^{\\pi/2}_0\\simplify{{a}*sin({b}x)*cos({c}x)}\\,dx\\]

", "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "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": "Integrate trig product 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": "

Consider the following integral:

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

", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..15)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(c+1..8)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "-{a}/({b}+{c})*sin(({b}+{c})*pi/3)+{a}/({b}-{c})*sin(({b}-{c})*pi/3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Select the relevant trigonometric product identity from the list below:

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Use the identity to rewrite $\\simplify{{a}*sin({b}x)*sin({c}x)}$:

You have not multiplied by $\\var{a}$.

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Hence write down the indefinite integral:

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

You forgot the constant of integration.

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Hence evaluate the definite integral:

\\[\\int_{-\\pi/3}^{\\pi/3}\\simplify{{a}*sin({b}x)*sin({c}x)}\\,dx\\]

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Completing the square works by noticing that

\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]

So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.

a)

Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.

\\begin{align}
\\simplify[basic]{x^2+{sml}x+{big}} &= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\
&= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}
\\end{align}

b)

We showed above that

\\[ \\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]

is equivalent to

\\[ \\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]

We can then rearrange this equation to solve for $x$.

\\begin{align}
\\simplify{(x+{bits[0]})^2-{(bits[1])^2} } &= 0 \\\\
(x+\\var{bits[0]})^2 &= \\var{bits[1]^2} \\\\
x+\\var{bits[0]} &= \\pm \\var{bits[1]} \\\\
x &= -\\var{bits[0]} \\pm \\var{bits[1]} \\\\[2em]

x_1 &= \\var{-bits[0]-bits[1]} \\text{,}\\\\
x_2 &= \\var{-bits[0]+bits[1]} \\text{.}
\\end{align}

", "variables": {"bits": {"description": "

After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.

", "definition": "sort(shuffle(1..9)[0..2])", "group": "Ungrouped variables", "name": "bits", "templateType": "anything"}, "big": {"description": "

The constant term in the expanded quadratic.

", "definition": "bits[0]^2-bits[1]^2", "group": "Ungrouped variables", "name": "big", "templateType": "anything"}, "sml": {"description": "

The coefficient of $x$ in the expanded quadratic.

", "definition": "2*bits[0]", "group": "Ungrouped variables", "name": "sml", "templateType": "anything"}}, "statement": "

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

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Write the following expression in the form $a(x+b)^2-c$.

$\\simplify {x^2+{sml}x+{big}} = $ [[0]]

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It doesn't look like you've completed the square.

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It doesn't look like you've completed the square.

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Now solve the quadratic equation

\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]

$x_1=$ [[0]]

$x_2=$ [[1]]

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Solve a quadratic equation by completing the square. The roots are not pretty!

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Completing the square works by noticing that

\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]

So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.

a)

Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.

\\begin{align}
\\simplify[basic]{x^2+{sml}x+{big}} &= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\
&= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}
\\end{align}

b)

We showed above that

\\[ \\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]

is equivalent to

\\[ \\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]

We can then rearrange this equation to solve for $x$.

", "variables": {"bits": {"description": "

After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.

", "definition": "sort(shuffle(1..9)[0..2])", "group": "Ungrouped variables", "name": "bits", "templateType": "anything"}, "big": {"description": "

The constant term in the expanded quadratic.

", "definition": "bits[0]^2-bits[1]^2", "group": "Ungrouped variables", "name": "big", "templateType": "anything"}, "sml": {"description": "

The coefficient of $x$ in the expanded quadratic.

", "definition": "2*bits[0]", "group": "Ungrouped variables", "name": "sml", "templateType": "anything"}}, "statement": "

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

Write the following expression in the form $a(x+b)^2-c$.

$\\simplify {x^2+{sml}x+{big}} = $ [[0]]

It doesn't look like you've completed the square.

Now solve the quadratic equation

\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]

$x_1=$ [[0]]

$x_2=$ [[1]]

Solve a quadratic equation by completing the square. The roots are not pretty!

Completing the square works by noticing that

\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]

So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.

a)

Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.

\\begin{align}
\\simplify[basic]{x^2+{sml}x+{big}} &= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\
&= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}
\\end{align}

b)

We showed above that

\\[ \\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]

is equivalent to

\\[ \\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]

We can then rearrange this equation to solve for $x$.

", "variables": {"bits": {"description": "

After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.

", "definition": "sort(shuffle(1..9)[0..2])", "group": "Ungrouped variables", "name": "bits", "templateType": "anything"}, "big": {"description": "

The constant term in the expanded quadratic.

", "definition": "bits[0]^2-bits[1]^2", "group": "Ungrouped variables", "name": "big", "templateType": "anything"}, "sml": {"description": "

The coefficient of $x$ in the expanded quadratic.

", "definition": "2*bits[0]", "group": "Ungrouped variables", "name": "sml", "templateType": "anything"}}, "statement": "

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

Write the following expression in the form $a(x+b)^2-c$.

$\\simplify {x^2+{sml}x+{big}} = $ [[0]]

It doesn't look like you've completed the square.

Now solve the quadratic equation

\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]

$x_1=$ [[0]]

$x_2=$ [[1]]

Solve a quadratic equation by completing the square. The roots are not pretty!

Completing the square works by noticing that

\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]

So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.

a)

Rewrite $x^2+\\var{sml}x$ as $\\simplify[basic]{ (x+{sml/2})^2 - {sml/2}^2}$.

\\begin{align}
\\simplify[basic]{x^2+{sml}x+{big}} &= \\simplify[basic]{(x+{sml/2})^2-{(sml/2)}^2+{big}} \\\\
&= \\simplify[basic]{(x+{sml/2})^2+{-(sml/2)^2+big}} \\text{.}
\\end{align}

b)

We showed above that

\\[ \\simplify[basic]{x^2+{sml}x+{big}} = 0 \\]

is equivalent to

\\[ \\simplify[basic]{(x+{bits[0]})^2-{bits[1]^2}} = 0 \\text{.} \\]

We can then rearrange this equation to solve for $x$.

", "variables": {"bits": {"description": "

After completing the square, the expression will have the form $(x + \\mathrm{bits}[0])^2 - \\mathrm{bits}[1]^2$.

", "definition": "sort(shuffle(1..9)[0..2])", "group": "Ungrouped variables", "name": "bits", "templateType": "anything"}, "big": {"description": "

The constant term in the expanded quadratic.

", "definition": "bits[0]^2-bits[1]^2", "group": "Ungrouped variables", "name": "big", "templateType": "anything"}, "sml": {"description": "

The coefficient of $x$ in the expanded quadratic.

", "definition": "2*bits[0]", "group": "Ungrouped variables", "name": "sml", "templateType": "anything"}}, "statement": "

We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a square plus another term - this is called \"completing the square\".

This can be useful when it isn't obvious how to fully factorise a quadratic equation.

Write the following expression in the form $a(x+b)^2-c$.

$\\simplify {x^2+{sml}x+{big}} = $ [[0]]

It doesn't look like you've completed the square.

Now solve the quadratic equation

\\[ \\simplify {x^2+{sml}x+{big}} = 0\\text{.} \\]

$x_1=$ [[0]]

$x_2=$ [[1]]

Solve a quadratic equation by completing the square. The roots are not pretty!

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

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