// Numbas version: exam_results_page_options
{"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Integration by parts", "tags": [], "metadata": {"description": "<p>Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx&nbsp;$ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n\t<p>Find the following indefinite integrals.</p>\n\t  <p>Input all numbers as fractions or integers and not decimals.</p>\n\t  <p>Input the constant of integration as $C$ where needed.</p>\n\t", "advice": "\n\t<p><strong>a)</strong></p>\n\t    <p>The formula for integrating by parts is<br/>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n\t    <p>We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.</p>\n\t    <p>So $\\displaystyle&nbsp;\\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.</p>\n\t    <p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \\int \\left( \\simplify[std]{(-{a}/{b})*cos({b}x+{c})}\\right) dx \\\\ &amp;=&amp; \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \\end{eqnarray} \\]</p>\n\t    <p><strong>b)</strong></p>\n\t    <p>For this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.</p>\n\t    <p>So $\\displaystyle&nbsp;\\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.</p>\n\t    <p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \\int \\left( \\simplify[std]{({a}/{b})*sin({b}x+{c})}\\right) dx \\\\ &amp;=&amp; \\simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \\end{eqnarray} \\]</p>\n\t    <p><strong>c)</strong></p>\n\t    <p>Using the results from Parts a and b, we have \\[\\begin{eqnarray*}I &amp;=&amp; \\int \\simplify[std]{{a1}x*sin({b}x+{c})} dx + \\int \\simplify[std]{{a2}x*cos({b}x+{c})} dx\\\\ &amp;=&amp; \\simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\\\ &amp;=&amp;\\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\\\ &amp;=&amp;\\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \\end{eqnarray*}\\]<br/>Hence <br/>$\\displaystyle&nbsp;\\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$</p>\n\t    <p>$\\displaystyle&nbsp;\\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$</p>\n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "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": "\n\t\t\t<p>$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]</p>\n\t\t\t      <p>Input all numbers as fractions or integers and not decimals.</p>\n\t\t\t      <p>Input the constant of integration as $C$.</p>\n\t\t\t      <p>You can get help by clicking on Show steps. You will lose 1 mark if you do.</p>\n\t\t\t", "stepsPenalty": 1, "steps": [{"type": "information", "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": "\n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t          <p>The formula for integrating by parts is<br/>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n\t\t\t\t\t          \n\t\t\t\t\t          \n\t\t\t\t\t"}], "gaps": [{"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, "answer": "({a}/{b^2})sin({b}x+{c}) - ({a}/{b})*x*cos({b}x+{c}) + C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>"}, "valuegenerators": [{"name": "c", "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": "\n\t\t\t<p>$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]</p>\n\t\t\t      <p>Input all numbers as fractions or integers and not decimals.</p>\n\t\t\t      <p>Input the constant of integration as $C$.</p>\n\t\t\t", "gaps": [{"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, "answer": "({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>"}, "valuegenerators": [{"name": "c", "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": "\n\t\t\t<p>Using the first two parts find:<br/>$\\displaystyle I=\\int \\simplify[std]{{a1}x*sin({b}x+{c})+{a2}x*cos({b}x+{c})} dx $<br/>You are given that \\[I=\\simplify[std]{f(x)*sin({b}x+{c})+g(x)*cos({b}x+{c})+C}\\]<br/>where $f(x)$ and $g(x)$ are polynomials of degree 1. 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