// Numbas version: exam_results_page_options
{"name": "Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "metadata": {"notes": "\n\t\t    \t\t<p><strong>2/08/2012:</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>Checked calculation. OK.</p>\n\t\t    \t\t<p>Added information about Show steps in prompt content area.&nbsp;</p>\n\t\t    \t\t<p>Got rid of a redundant ruleset.&nbsp;!noLeadingMinus added to std ruleset.</p>\n\t\t    \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$</p>"}, "showQuestionGroupNames": false, "advice": "\n\t    \n\t    \n\t    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a}+{b}cos(x)}$ then $du=\\simplify[std]{({-b}*sin(x))*dx }$<br/>Hence we can replace $\\sin(x)\\;dx$ by $\\frac{1}{\\var{-b}}\\;du$.</p>\n\t    \n\t    \n\t    \n\t    <p>Hence the integral becomes:</p>\n\t    \n\t    \n\t    \n\t    <p>\\[\\begin{eqnarray*} I&#38;=&#38;\\simplify[std]{Int((1/{-b})u^{m},u)}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n\t    \n\t    \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t  \n\t", "functions": {}, "progress": "ready", "tags": ["Calculus", "Steps", "calculus", "constant of integration", "integrals", "integrating trigonometric functions", "integration", "integration by substitution", "steps", "substitution"], "variables": {"b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(3..9)", "name": "m"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "a": {"definition": "random(1..9)", "name": "a"}}, "name": "Julie's copy of Indefinite integral by substitution", "parts": [{"prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]</p>\n\t\t\t        <p>Input all numbers as integers or fractions.</p>\n\t\t\t        <p>$I=\\;$[[0]]</p>\n\t\t\t        <p>Input the constant of integration as $C$.</p>\n\t\t\t        <p>Click on Show steps if you need help. You will lose 1 mark if you do so.</p>\n\t\t\t      \n\t\t\t", "stepspenalty": 1.0, "marks": 0.0, "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$</p>", "marks": 0.0, "type": "information"}], "type": "gapfill", "gaps": [{"notallowed": {"strings": ["."], "partialcredit": 0.0, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions.</p>", "showstrings": false}, "answer": "({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C", "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "marks": 3.0, "answersimplification": "std", "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme"}]}], "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "name": "", "questions": []}], "type": "question", "extensions": [], "statement": "\n\t  <p>Find the following integral.</p>\n\t    <p>Input the constant of integration as $C$.</p>\n\t    <p>Input all numbers as integers or fractions not as decimals.</p>\n\t  \n\t", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}