// Numbas version: exam_results_page_options
{"name": "Integration 3 - Substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c", "p"], "name": "Integration 3 - Substitution", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>If you do not understand how to begin these questions, please see&nbsp;<em>'Integration 1 - Substitution'</em>.</p>\n<p>The difference with these questions is that you must rearrange the $u=$ equation for $x$ and put&nbsp;this in after the $dx$ substitution as it does not nicely cancel as it did before.</p>\n<p>Once you have done this, it should be clear that you then multiply through the brackets and integrate the two remaining terms with respect to $u$.</p>\n<p>Remember to replace the '$u$'s at the end.</p>", "rulesets": {}, "parts": [{"prompt": "<p>Use $u=\\simplify{{c[1]}x+{c[2]}}$ as your substitution.</p>\n<p>$\\simplify{Int({c[0]}x*({c[1]}x+{c[2]})^{p[0]},x)}$</p>\n<p>$=\\int$ [[1]]&nbsp;$du$</p>\n<p>$=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all, simplifyFractions, otherNumbers", "scripts": {}, "answer": "({c[0]}/{c[1]}^2)*(1/({p[0]}+2)*({c[1]}x+{c[2]})^({p[0]}+2)-{c[2]}/({p[0]}+1)*({c[1]}x+{c[2]})^({p[0]}+1))", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[0]}/{c[1]}^2)(u-{c[2]})(u^{p[0]})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Use $u=\\simplify{{c[4]}x+{c[5]}}$ as your substitution.</p>\n<p>$\\simplify{Int({c[3]}x*({c[4]}x+{c[5]})^{p[1]},x)}$</p>\n<p>$=\\int$ [[1]]&nbsp;$du$</p>\n<p>$=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[3]}/{c[4]}^2)*(1/({p[1]}+2)*({c[4]}x+{c[5]})^({p[1]}+2)-{c[5]}/({p[1]}+1)*({c[4]}x+{c[5]})^({p[1]}+1))", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[3]}/{c[4]}^2)(u-{c[5]})(u^{p[1]})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p><strong>Integrate the following by substitution.</strong></p>\n<p>You may assume the constant of integration is zero for the purposes of these questions.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"p": {"definition": "repeat(random(2..5),5)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "c": {"definition": "repeat(random(-5..5 except 0 except 1 except -1),9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}}, "metadata": {"notes": "", "description": "<p>Solving integration by substitution and rearranging $u$ for $x$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}]}], "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}]}