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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\\]

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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\\]

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The following integral can be evaluated by using substitution:

\n

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

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

\n

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

\n

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

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

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

\n

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

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Hence write down the indefinite integral:

\n

\\[\\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.

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

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

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Evaluate the definite integral:

\n

\\[\\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:

\n

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

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

\n

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

\n

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]]
\n

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

\n

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

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Hence write down the indefinite integral:

\n

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

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

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

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

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

\n

\\[\\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:

\n

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

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

\n

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

\n

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]]$)$
\n

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

\n

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

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Hence write down the indefinite integral:

\n

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

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

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

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

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

\n

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

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Consider the following integral:

\n

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

\n

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 $u\\frac{dv}{dv}$.

\n

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]]$dx$
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Hence write down the indefinite integral:

\n

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

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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:

\n

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

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The following integral may be evaluated by using integration by parts.

\n

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

\n

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

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

\n

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]]
", "gaps": [{"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, "answer": "{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": ""}]}, {"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, "answer": "{a}", "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": []}, {"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, "answer": "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": ""}]}, {"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, "answer": "exp({b}x)/{b}", "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": ""}]}], "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": "

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]]$dx$
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Hence write down the indefinite integral:

\n

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

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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:

\n

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

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The following integral may be evaluated by using integration by parts.

\n

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

\n

Notes:

\n", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-8..8 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(0 .. 4#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "{a}*2^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}*2)-1)-{a}*1/({n}+1)^2*(({n}+1)*ln({b})-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "ans", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "

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

\n

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]]
", "gaps": [{"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, "answer": "ln({b}*x)", "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": "x", "value": ""}]}, {"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, "answer": "1/x", "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": "x", "value": ""}]}, {"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, "answer": "{a}*x^{n}", "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": "x", "value": ""}]}, {"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, "answer": "{a}*x^({n}+1)/({n}+1)", "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": "x", "value": ""}]}], "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": "

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]]$dx$
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Hence write down the indefinite integral:

\n

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

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

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Evaluate the definite integral:

\n

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

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The following integral can be integrated using partial fractions:

\n

\\[\\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]]

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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})}$
", "gaps": [{"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, "answer": "A*(x+{d})+B*(x+{c})", "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": ""}, {"name": "b", "value": ""}, {"name": "x", "value": ""}]}], "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": "

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):

\n

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

\n

", "gaps": [{"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": "{a+b}*x+{a*d+b*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": "x", "value": ""}]}, {"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": "A*(x+{d})+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": "a", "value": ""}, {"name": "b", "value": ""}, {"name": "x", "value": ""}]}], "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 $x=\\var{-c}$ into the numerators: [[0]] = [[1]]

\n

Therefore $A=$ [[2]]

", "gaps": [{"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": "{xmc}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"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": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "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": "a", "maxValue": "a", "correctAnswerFraction": false, "allowFractions": false, "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]]

\n

Therefore $B=$ [[2]]

", "gaps": [{"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": "{xmd}", "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": []}, {"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": "({c}-{d})*B", "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": "b", "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": "b", "maxValue": "b", "correctAnswerFraction": false, "allowFractions": false, "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": "

Therefore 

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

\n

\\[\\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:

\n

\\[\\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}}}\\]

"], "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": "

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})}$
", "gaps": [{"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, "answer": "A*(x+{d})+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": "a", "value": ""}, {"name": "b", "value": ""}, {"name": "x", "value": ""}]}], "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 $x=\\var{-c}$ into the numerators above: [[0]] = [[1]]

\n

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]]

\n

Therefore $B=$ [[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*d", "maxValue": "b-a*d", "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": "({c}-{d})*B", "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": "b", "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": "n2", "maxValue": "n2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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 calculate the following indefinite integral:

\n

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

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

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:

\n

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

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The following integral involves a trigonometric product:

\n

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

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Select the relevant trigonometric product identity from the list below:

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

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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:

\n

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

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

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Hence evaluate the definite integral:

\n

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

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Consider the following integral:

\n

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

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Select the relevant trigonometric product identity from the list below:

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

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

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Hence write down the indefinite integral:

\n

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

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You forgot the constant of integration.

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Hence evaluate the definite integral:

\n

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

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