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{"name": "Ch1: Integration by parts ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["s3", "c", "b", "m"], "name": "Ch1: Integration by parts ", "tags": ["algebraic manipulation", "Calculus", "calculus", "indefinite integration", "integrals", "integration", "integration by parts", "steps", "Steps"], "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 $u = x$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{({b}*x+{c})^{m}}$.</p>\n<p>So $\\displaystyle \\frac{du}{dx}$ = $1$ and $\\displaystyle v = \\simplify[std]{(1/{(m+1)*b})*({b}*x+{c})^{m+1}}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray*} \\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*b})*Int (({b}*x+{c})^{m+1}, x)} \\\\ &amp;=&amp; \\simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*(m+2)*b^2})*({b}*x+{c})^{m+2}+C} \\\\ &amp;=&amp;\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+2)}x - ({b}x+{c}))+C}\\\\ &amp;=&amp;\\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}x - {c})+C} \\end{eqnarray*}\\]<br/>The solution is: $\\simplify[std]{g(x)={b*(m+1)}*x-{c}}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": "0", "prompt": "<p>$I=\\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx $<br/><br/></p>\n<p>Leave your answer in the form of $\\frac{(ax+b)^d}{e}*g(x) + C$ with your constant of integration as \"+ C\".</p>\n<p>$I=\\;$[[0]]</p>\n<p></p>\n<p>You can get help by clicking on show steps. </p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({b}x+{c})^{m+1}/{b^2*(m+1)*(m+2)}*({b*(m+1)}*x-{c})+C", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "steps": [{"prompt": "<p>The formula for integrating by parts is<br/>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]<br/><br/></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p>Find the following indefinite integral.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "m": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "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>Checked calculation. OK.</p>\n    \t\t    \t\t<p>Got rid of redundant instructions about inputting constant of integration.</p>\n    \t\t    \t\t<p>Got rid of instruction re not inputting decimals - no restriction needed, so no forbidden strings.</p>\n    \t\t    \t\t<p>Penalised use of steps, 1 mark. Added message to that effect.</p>\n    \t\t    \t\t<p>Improved Advice display.</p>\n    \t\t    \n    \t\t", "description": "<p>Given that $\\displaystyle \\int x({ax+b)^{m}} dx=\\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Graham Wynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/602/"}]}]}], "contributors": [{"name": "Graham Wynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/602/"}]}