// Numbas version: exam_results_page_options {"showstudentname": true, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Basic integration 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "preamble": {"js": "", "css": ""}, "variables": {"c": {"definition": "random(-5..5 except [-1,0])", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "c"}, "b": {"definition": "random(2..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b"}, "d": {"definition": "random(1..15)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "d"}, "a": {"definition": "random(1..8)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a"}}, "variable_groups": [], "parts": [{"prompt": "

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

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

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

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\$\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx\$

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

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It is possible to write it without a fraction.

", "strings": ["/"], "showStrings": true}, "checkingtype": "absdiff", "showCorrectAnswer": true, "scripts": {}, "answer": "{c}*x^-2", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showpreview": true}], "showCorrectAnswer": true, "scripts": {}, "answer": "{-c}/x+{1/(a+1)}*x^{b+1}+c", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "vsetrangepoints": 5, "showpreview": true}, {"prompt": "

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

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Integrate each of the following basic functions indefinitely.

\n

Don't forget the constant of integration.

", "ungrouped_variables": ["a", "b", "c", "d"], "advice": "", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question"}, {"name": "Basic integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "rulesets": {}, "metadata": {"description": "", "licence": "None specified"}, "ungrouped_variables": ["a", "b", "c", "a1", "d", "a2", "a3", "a4", "a5"], "tags": [], "statement": "

Integrate each of the following functions with the given limits.

\n

", "parts": [{"prompt": "

Integrate and evaluate

\n

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

", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a1}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "

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\$

", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "-{b}/{c+1}*x^{-c-1}+{2*d}/3*x^(3/2)", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "

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\$

", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a*(c+1)}*exp(x/{c+1})+{d}*x", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "

Integrate and evaluate

\n

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

", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a4}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "

First integrate indefinitely

\n

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

", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{d}/{a}*ln(x)+{a}/{d*c}*sin({d}*x)", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}, {"prompt": "

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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", "templateType": "anything", "definition": "(a*(c+1)*exp(b/(c+1))+d*b)-(a*(c+1))", "group": "Ungrouped variables"}}, "type": "question"}, {"name": "Integration by parts", "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}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "

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]] $dx$
", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x*sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "

Hence write down the indefinite integral:

\n

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

\n

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

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

", "showStrings": false, "strings": ["+c"], "partialCredit": 0}, "scripts": {}, "answer": "{a/b}*x*sin({b}*x)+{a/b^2}*cos({b}*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral:

\n

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

", "expectedvariablenames": [], "checkingaccuracy": "2", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "

Consider the following integral:

\n

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

\n

This may be evaluated by using integration by parts.

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..3#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}/{b}*sin({b})+{a}/{b}^2*(cos({b})-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "n", "b", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "

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

\n

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

\n

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

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "x^({n}+1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "du/(({n}+1)*x^{n})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n\n\n\n\n\n\n\n\n\n\n
 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]]

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}*sin(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({n}+1)*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/({n}+1)*sin(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/({n}+1)*cos(u)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

Hence write down the indefinite integral:

\n

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

\n

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

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/({n}+1)*cos(x^({n}+1)+{b})+c", "steps": [{"showCorrectAnswer": true, "prompt": "

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

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Evaluate the definite integral, correct to 1 decimal place:

\n

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

", "expectedvariablenames": [], "checkingaccuracy": "1", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "dp", "checkvariablenames": false, "type": "jme"}], "statement": "

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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First test

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