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

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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\$

", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a}*x^2/2+1/{b}*cos({b}*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{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\$

", "variableReplacements": [], "vsetrangepoints": 5, "marks": "2", "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "{a3}", "checkingtype": "sigfig", "type": "jme", "stepsPenalty": 0, "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrange": [0, 1], "checkingaccuracy": 3, "steps": [{"prompt": "

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\$

", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "scripts": {}, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "answer": "x^{a+1}/{b*(a+1)}-{c}/{b}*exp({-b}*x)-{d}*x", "checkingtype": "absdiff", "showpreview": true, "showCorrectAnswer": true, "vsetrange": [0, 1], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme"}], "showpreview": false}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "advice": "", "variables": {"d": {"name": "d", "description": "", "templateType": "anything", "definition": "random(-5..5 except 0)", "group": "Ungrouped variables"}, "a2": {"name": "a2", "description": "

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", "templateType": "anything", "definition": "(d/a*ln(c+1)+a/(d*c)*sin(d*(c+1)))-(d/a*ln(c)+a/(d*c)*sin(d*c))", "group": "Ungrouped variables"}, "a5": {"name": "a5", "description": "

", "templateType": "anything", "definition": "(c^(a+1)/(b*(a+1))-c/b*exp(-b*c)-d*c)-(-c/b)", "group": "Ungrouped variables"}, "b": {"name": "b", "description": "", "templateType": "anything", "definition": "random(1..9 except a)", "group": "Ungrouped variables"}, "a": {"name": "a", "description": "", "templateType": "anything", "definition": "random(1..9)", "group": "Ungrouped variables"}, "c": {"name": "c", "description": "", "templateType": "anything", "definition": "random(1..5)", "group": "Ungrouped variables"}, "a3": {"name": "a3", "description": "

", "templateType": "anything", "definition": "(a*(c+1)*exp(b/(c+1))+d*b)-(a*(c+1))", "group": "Ungrouped variables"}}, "type": "question"}]}, {"pickQuestions": 1, "name": "Substitution", "pickingStrategy": "all-ordered", "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]]

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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.

", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

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\$

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..8#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-5..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "-{a}/({n}+1)*cos(pi^({n}+1)+{b})+{a}/({n}+1)*cos({b})", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n": {"definition": "random(1..4#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by substitution 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/"}], "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\n\n\n\n
 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]]

Hence write down the indefinite integral:

\n

\$\\int\\simplify{{a}*x^{n}/sqrt(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": "2*{a}/({n}+1)*sqrt(x^({n}+1)+{b})+c", "steps": [{"showCorrectAnswer": true, "prompt": "

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

", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to the nearest whole number:

\n

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

", "expectedvariablenames": [], "checkingaccuracy": "0", "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}/sqrt(x^({n}+1)+{b})}\\,dx\$

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(1..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "2*{a}/({n}+1)*(sqrt(10^({n}+1)+{b})-sqrt({b}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n": {"definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by substitution 3", "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", "m"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "

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

\n

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

\n

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

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "x^{m}+{m}*{b}*x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{m}*x^({m}-1)+{m}*{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{m}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "x^({m}-1)+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n\n\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]] $($[[2]]$)$
\n

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

\n

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

Hence write down the indefinite integral:

\n

\$\\int\\simplify{{a}(x^({m}-1)+{b})(x^{m}+{m}*{b}*x)^{n}}\\,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}/({m}*({n}+1))*(x^{m}+{m}*{b}*x)^({n}+1)+c", "steps": [{"showCorrectAnswer": true, "prompt": "

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

", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to 4 significant figures:

\n

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

", "expectedvariablenames": [], "checkingaccuracy": "4", "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "

The following integral can be evaluated by using substitution:

\n

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

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "ans": {"definition": "{a}/({m}*({n}+1))*((1+{m}*{b})^({n}+1))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "b": {"definition": "random(-6..6 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "m": {"definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "m", "description": ""}, "n": {"definition": "random(3..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"pickQuestions": 1, "name": "Parts", "pickingStrategy": "all-ordered", "questions": [{"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]]
", "marks": 0, "gaps": [{"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "cos({b}x)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "sin({b}x)/{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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]] $dx$
", "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.)

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "2", "showCorrectAnswer": true, "musthave": {"message": "

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

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

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

Evaluate the definite integral, correct to 2 decimal places:

\n

\$\\int_{-1}^1\\simplify{{a}x*exp({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": "

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($\\simplify{{b}*x}$).

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-8..8 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "b": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}/{b}*(exp({b})+exp(-{b}))-{a}/{b}^2*(exp({b})-exp(-{b}))", "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 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}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "ans", "n"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"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]]
", "marks": 0, "gaps": [{"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "ln({b}*x)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": "5", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "1/x", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^({n}+1)/({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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]] $dx$
", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^({n}+1)*ln({b}x)/({n}+1)", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*x^{n}/({n}+1)", "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^{n}*ln({b}*x)}\\,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, "musthave": {"message": "

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

", "showStrings": false, "strings": ["+c"], "partialCredit": 0}, "scripts": {}, "answer": "{a}*x^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}x)-1)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to 3 significant figures:

\n

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

", "expectedvariablenames": [], "checkingaccuracy": 3, "vsetrange": [0, 1], "showpreview": false, "marks": "2", "showCorrectAnswer": true, "scripts": {}, "answer": "{ans}", "checkingtype": "sigfig", "checkvariablenames": false, "type": "jme"}], "statement": "

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

\n

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

\n

Notes:

\n
\n
• powers can be entered using ^
• \n
• to divide, use /
• \n
", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-8..8 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "n": {"definition": "random(0..4#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "n", "description": ""}, "b": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "{a}*2^({n}+1)/({n}+1)^2*(({n}+1)*ln({b}*2)-1)-{a}*1/({n}+1)^2*(({n}+1)*ln({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": []}]}]}, {"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}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "xmc", "xmd"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "

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

", "marks": 0, "gaps": [{"notallowed": {"message": "

You should factorise and will not need to use powers.

", "showStrings": false, "strings": ["^"], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "(x+{c})*(x+{d})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"maxAnswers": 0, "displayColumns": 0, "prompt": "

Select the correct form of the partial fraction:

", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "minAnswers": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "maxMarks": 0, "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}}}\$

"], "type": "1_n_2", "distractors": ["", "", "", ""], "minMarks": 0}, {"prompt": "

Cross-multiply: (Note:You will need to use * for multiplication.)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
 $A$ $+$ $B$ $=$ [[0]] $\\simplify{x+{c}}$ $\\simplify{x+{d}}$ $\\simplify{(x+{c})*(x+{d})}$
", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a+b}*x+{a*d+b*c}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

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

\n

Therefore $A=$ [[2]]

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{xmc}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({d}-{c})*A", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

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

\n

Therefore $B=$ [[2]]

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{xmd}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}-{d})*B", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"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]] $\\simplify[all]{x^2+{c+d}*x+{c*d}}$ $\\simplify{x+{c}}$ $\\simplify{x+{d}}$
", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "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\$

\n

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

", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{a}*ln(x+{c})+{b}*ln(x+{d})+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "statement": "

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\$

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(-4..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-4..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-4..4 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-4..4 except c)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "xmd": {"definition": "-{b}*{d}+{b}*{c}", "templateType": "anything", "group": "Ungrouped variables", "name": "xmd", "description": ""}, "xmc": {"definition": "-{a}*c+{a}*{d}", "templateType": "anything", "group": "Ungrouped variables", "name": "xmc", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integrate with partial fractions 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/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "ans", "n1", "n2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"maxAnswers": 0, "displayColumns": 0, "prompt": "

Select the correct form of the partial fraction:

", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "marks": 0, "minAnswers": 0, "displayType": "radiogroup", "showCorrectAnswer": true, "scripts": {}, "maxMarks": 0, "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}}}\$

"], "type": "1_n_2", "distractors": ["", "", "", ""], "minMarks": 0}, {"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]] $\\simplify{(x+{c})*(x+{d})}$ $\\simplify{x+{c}}$ $\\simplify{x+{d}}$ $\\simplify{(x+{c})*(x+{d})}$
", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "A*(x+{d})+B*(x+{c})", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

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

\n

Therefore $A=$ [[2]]

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b-a*c}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({d}-{c})*A", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{n1}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "

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

\n

Therefore $B=$ [[2]]

", "marks": 0, "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "{b-a*d}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": false, "marks": "0.5", "showCorrectAnswer": true, "scripts": {}, "answer": "({c}-{d})*B", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "answer": "{n2}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"vsetrangepoints": 5, "prompt": "

Hence calculate the following indefinite integral:

\n

\$\\int\\frac{\\simplify[all]{{a}*x+{b}}}{\\simplify{(x+{c})*(x+{d})}}\\,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": "{n1}*ln(x+{c})+{n2}*ln(x+{d})+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to 2 significant figures:

\n

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

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

The following integral can be integrated using partial fractions:

\n

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

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": "15", "condition": ""}, "variables": {"a": {"definition": "random(-5..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(0..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-5..5 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "c+random(3,4,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "ans": {"definition": "{n1}*(ln(10+{c})-ln(1+{c}))+{n2}*(ln(10+{d})-ln(1+{d}))", "templateType": "anything", "group": "Ungrouped variables", "name": "ans", "description": ""}, "n1": {"definition": "(b-a*c)/(d-c)", "templateType": "anything", "group": "Ungrouped variables", "name": "n1", "description": ""}, "n2": {"definition": "(b-a*d)/(c-d)", "templateType": "anything", "group": "Ungrouped variables", "name": "n2", "description": ""}}, "metadata": {"notes": "", "description": ""}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"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}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

Rewrite $\\simplify{{a}*sin({b}x)*cos({c}x)}$ using one of the trigonometric product identities:

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

The 4 trigonometric product identities are:

\n

\\begin{eqnarray}\\sin(A)\\cos(B)&&=\\frac{1}{2}[\\sin(A+B)+\\sin(A-B)]\\\\
\\cos(A)\\sin(B)&&=\\frac{1}{2}[\\sin(A+B)-\\sin(A-B)]\\\\
\\cos(A)\\cos(B)&&=\\frac{1}{2}[\\cos(A+B)+\\cos(A-B)]\\\\
\\sin(A)\\sin(B)&&=-\\frac{1}{2}[\\cos(A+B)-\\cos(A-B)]\\end{eqnarray}

", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Hence write down the indefinite integral:

\n

\$\\int\\simplify{{a}*sin({b}x)*cos({c}x)}\\,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}/(2({b}+{c}))*cos(({b}+{c})*x)-{a}/(2({b}-{c}))*cos(({b}-{c})*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to 1 decimal place:

\n

\$\\int^{\\pi/2}_0\\simplify{{a}*sin({b}x)*cos({c}x)}\\,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 involves a trigonometric product:

\n

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

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "2*random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(c+1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"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}))", "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": "Integrate trig product 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/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "ans"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "

Rewrite $\\simplify{{a}*sin({b}x)*sin({c}x)}$ using one of the trigonometric product identities:

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

The 4 trigonometric product identities are:

\n

\\begin{eqnarray}\\sin(A)\\cos(B)&&=\\frac{1}{2}[\\sin(A+B)+\\sin(A-B)]\\\\
\\cos(A)\\sin(B)&&=\\frac{1}{2}[\\sin(A+B)-\\sin(A-B)]\\\\
\\cos(A)\\cos(B)&&=\\frac{1}{2}[\\cos(A+B)+\\cos(A-B)]\\\\
\\sin(A)\\sin(B)&&=-\\frac{1}{2}[\\cos(A+B)-\\cos(A-B)]\\end{eqnarray}

", "scripts": {}, "type": "information", "marks": 0}], "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Hence write down the indefinite integral:

\n

\$\\int\\simplify{{a}*sin({b}x)*sin({c}x)}\\,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}/(2({b}+{c}))*sin(({b}+{c})*x)+{a}/(2({b}-{c}))*sin(({b}-{c})*x)+c", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme"}, {"vsetrangepoints": 5, "prompt": "

Evaluate the definite integral, correct to 2 significant figures:

\n

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

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

The following integral involves a trigonometric product:

\n

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

", "variable_groups": [], "progress": "in-progress", "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(3..15)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(c+1..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "ans": {"definition": "-{a}/({b}+{c})*sin(({b}+{c})*pi/3)+{a}/({b}-{c})*sin(({b}-{c})*pi/3)", "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": "Blathnaid's copy of Integration revision", "type": "exam", "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "Rob Beckett", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2450/"}], "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"], ["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"], ["question-resources/undefined_14", "/srv/numbas/media/question-resources/undefined_14"], ["question-resources/undefined_15", "/srv/numbas/media/question-resources/undefined_15"]]}