// 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": [{"variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "name": "c"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "name": "a1"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "name": "a2"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "t"}}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Integration By Parts", "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "- ({a}/{b})*x*({t}*cos({b}x+{c})-{1-t}*exp({b}x+{c})) +({a}/{b^2})({t}*sin({b}x+{c})-{1-t}*exp({b}x+{c})) + C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "<p>$\\displaystyle \\int \\simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c}))} dx = \\phantom{{}}$[[0]]</p>", "steps": [{"type": "information", "prompt": "\n          \n          \n          <p>The formula for integrating by parts is<br/>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n          \n          \n            \n            ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the following indefinite integral.</p>\n  <p>Input all numbers as fractions or integers and not decimals.</p>\n  <p>You must input the constant of integration as $C$.</p>\n  <p>Note that if you are entering an expression such as $x*\\cos(p)$ for $p$ some expression then you must enter it as x*cos(p).</p>", "tags": ["Calculus", "calculus", "checked2015", "exponential function", "indefinite integration", "integration", "integration by parts", "integration of trigonometric functions", "mas1601", "MAS1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n                                \t\t        \t\t<p><strong>20/06/2012:</strong></p>\n                                \t\t        \t\t<p>Added tags.</p>\n                                \t\t        \t\t<p>Changed xsin to x*sin in Advice.</p>\n                                \t\t        \t\t<p>Tidied up display. Checked calculations. Users have to include capital C for the first two questions.</p>\n                                \t\t        \t\t<p><strong>4/07/2012:<br /></strong></p>\n                                \t\t        \t\t<p>Edited part c to include brackets around the integrand.<strong><br /></strong></p>\n                                \t\t        \t\t<p>Added tags.</p>\n                                \t\t        \t\t<p><strong>9/07/2012:</strong></p>\n                                \t\t        \t\t<p>Added !noLeadingMinus to ruleset so that integrating by parts expressions are in the expected order.</p>\n                                \t\t        \t\t<p>Improved display of last two lines of Advice.</p>\n                                \t\t        \n                                \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Integrating by parts. </p>\n<p>Find $ \\int ax\\sin(bx+c)\\;dx$ or $\\int ax e^{bx+c}\\;dx$&nbsp;</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The formula for integrating by parts is</p>\n  <p>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n  <p>We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{{t}*sin({b}x+{c})+{1-t}*exp({b}x+{c})}$.</p>\n  <p>So $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $v = \\simplify[std]{({-t}/{b})*cos({b}x+{c})+{1-t}/{b}*exp({b}x+{c})}$.</p>\n  <p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*({t}sin({b}x+{c})+{1-t}*exp({b}x+{c}))&nbsp;} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}))-{a}/{b}int (({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}),x)} \\\\ &amp;=&amp;&nbsp; \\simplify[std]{({a}/{b})*x*(({-t})*cos({b}x+{c})+{1-t}*exp({b}x+{c}))&nbsp;&nbsp;+ ({t*a}/{b^2})*sin({b}x+{c}) -{(1-t)*a}/{b^2}*exp({b}x+{c})+ C} \\end{eqnarray} \\]</p>\n  <p><strong>&nbsp;</strong></p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}