// Numbas version: exam_results_page_options
{"name": "Integration by parts 2 - cos and sin with limits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "i1", "i0"], "name": "Integration by parts 2 - cos and sin with limits", "tags": ["Calculus", "calculus", "constant of integration", "indefinite integration", "integrals", "integrating", "integrating trigonometric functions", "integration by parts", "steps", "Steps", "twice"], "advice": "<h4>The formula for integrating by parts is</h4>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n<p>We choose $u = \\simplify[std]{({a}x+{b})}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{cos({c}*x+{d})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx} = \\simplify[std]{{a}}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{c})*sin({c}*x+{d})}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} \\simplify[std]{({a}*x+{b})*cos({c}*x+{d})} dx &amp;=&amp; [uv]^\\simplify{{i1}/2}_\\simplify{{i0}/2} - \\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} v \\frac{du}{dx} dx \\\\ &amp;=&amp; [\\simplify[std]{(({a}*x+{b})/{c})*sin({c}*x+{d})}]^\\simplify{{i1}/2}_\\simplify{{i0}/2}&nbsp;- \\simplify{({a}/{c})}\\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} \\sin({c}*x+{d}) \\mathrm{d}x \\\\ &amp;=&amp; [\\simplify[std]{(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d})}]^\\simplify{{i1}/2}_\\simplify{{i0}/2}\\end{eqnarray} \\]</p>\n<h4></h4>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>$I=\\displaystyle \\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx $</p>\n<p>The formula for integration by parts is</p>\n<p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n<p>What is the most suitable choice for $u$ and $\\frac{dv}{dx}$?</p>\n<p>$u =\\;$[[0]]</p>\n<p>$\\frac{dv}{dx} =\\;$[[1]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{a}x+{b}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "cos({c}x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence find $\\frac{du}{dx} =\\;$[[0]]</p>\n<p>$v =\\;$[[1]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "1/{c}sin({c}x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Hence find $uv =\\;$[[0]]</p>\n<p>$\\int v\\frac{du}{dx}\\mathrm{d}x = \\;$[[1]]$+C$</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "({a}x+{b})/{c}sin({c}x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "-{a}/{c}^2cos({c}x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Use the results from above&nbsp;to find:</p>\n<p>$I=\\displaystyle \\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx =&nbsp;\\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} u\\frac{dv}{dx} dx = [uv]^\\simplify{{i1}/2}_\\simplify{{i0}/2}- \\int^\\simplify{{i1}/2}_\\simplify{{i0}/2} v \\frac{du}{dx} dx = \\;$[[0]]</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "({a}{i1}/2+{b})/{c}sin({c}{i1}/2+{d})+{a}/{c}^2cos({c}{i1}/2+{d})-({a}{i0}/2+{b})/{c}sin({c}{i0}/2+{d})-{a}/{c}^2cos({c}{i0}/2+{d})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Find the following indefinite integral.</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p>Input the constant of integration as $C$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "i1>i0"}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "0", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "i1": {"definition": "random(0,pi,2pi,3pi)", "templateType": "anything", "group": "Ungrouped variables", "name": "i1", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "i0": {"definition": "random(0,pi,2pi,3pi)", "templateType": "anything", "group": "Ungrouped variables", "name": "i0", "description": ""}}, "metadata": {"notes": "\n    \t\t    \t\t<p><strong>3/08/2012:</strong></p>\n    \t\t    \t\t<p>Added tags.</p>\n    \t\t    \t\t<p>Added description.</p>\n    \t\t    \t\t<p>Got rid of redundant ruleset, added !noLeadingMinus to std ruleset as we need to keep the standard order for integrating by parts.</p>\n    \t\t    \t\t<p>Checked calculation. OK.</p>\n    \t\t    \t\t<p>Penalised use of steps in first part, 1 mark. Added message to that effect in first part.</p>\n    \t\t    \t\t<p>Added message about not inputting decimals in appropriate places.</p>\n    \t\t    \t\t<p>Changed marks reflecting the use of steps and degree of difficulty in second part.</p>\n    \t\t    \t\t<p>Improved Advice display.</p>\n    \t\t    \n    \t\t", "description": "<p>Find $\\displaystyle \\int (ax+b)\\cos(cx+d)\\; dx $</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}