// Numbas version: exam_results_page_options
{"name": "Phil's copy of Integration by parts, Exponential 20/10", "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", "s3", "s2", "s1", "a1", "a2"], "name": "Phil's copy of Integration by parts, Exponential 20/10", "tags": ["algebraic manipulation", "Calculus", "calculus", "exponential function", "integrals", "integration", "integration by parts", "integration of exponential function", "steps", "Steps"], "advice": "<h4>a)</h4>\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<p>We choose $u = \\simplify[std]{{a}x+{b}}$ and $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx} = \\var{a}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{c})*e^({c}*x)}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*e^({c}*x)} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{({1}/{c})*({a}x+{b})*e^({c}x) - (1/{c})*Int(({a})*e^({c}x),x)} \\\\ &amp;=&amp; \\simplify[std]{(1/{c})*({a}x+{b})*e^({c}x) -({a}/{c^2})*e^({c}x) + C}\\\\ &amp;=&amp; \\simplify[std]{(({a}x+{b})/{c}-{a}/{c^2})*e^({c}*x) + C}\\\\ &amp;=&amp; \\simplify[std]{(({a}/{c})x+{b*c-a}/{c^2})*e^({c}*x) + C} \\end{eqnarray} \\]</p>\n<p>Hence $\\displaystyle&nbsp;\\simplify[std]{g(x)=({a}/{c})*x+{c*b-a}/{c^2}}$</p>\n<h4>b)</h4>\n<p>For this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle&nbsp;\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx}$ = $\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{c})*e^({c}*x)}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{({1}/{c})*({a}x+{b})^2*e^({c}x) - (1/{c})*Int({2*a}*({a}x+{b})*e^({c}x),x)} \\\\ &amp;=&amp; \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*Int(({a}x+{b})*e^({c}x),x)}\\dots (*) \\end{eqnarray*}\\]</p>\n<p>But in Part a we have aready worked out $\\displaystyle&nbsp;\\simplify[std]{Int(({a}x+{b})*e^({c}*x),x)=(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}$&nbsp;</p>\n<p>So on substituting this in equation (*) we find:<br/>\\[ \\begin{eqnarray*}I&amp;=&amp; \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}\\\\ &amp;=&amp; \\simplify[std]{({a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3})*e^({c}x) +C} \\end{eqnarray*}\\]</p>\n<p>Hence $\\displaystyle&nbsp;\\simplify[std]{h(x)={a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}}$</p>\n<p>More information on the integration of exponentials is in chapter 3 of your Mathematical Physics book, section 3.8 on page 73-75.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": "0", "prompt": "<p>$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*e^({c}x)} dx $<br/>You are given that the answer is of the form \\[I=g(x)e^{\\var{c}x}+C\\] for a polynomial $g(x)$.&nbsp;You have to find $g(x)$.</p>\n<p>$g(x)=\\;$[[0]]</p>\n<p>Input all numbers as fractions or integers and not decimals.</p>\n<p>You can get help by clicking on Show steps. You will not lose any marks if you do.</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", "scripts": {}, "answer": "({a}/{c})*x+{c*b-a}/{c^2}", "marks": 2, "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. \\]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"prompt": "<p>Use the result from the first part to find:</p>\n<p>$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*e^({c}x)} dx $</p>\n<p>You are given that the answer is of the form \\[I=h(x)e^{\\var{c}x}+C\\] for a polynomial $h(x)$.&nbsp;You have to find $h(x)$.</p>\n<p>$h(x)=\\;$[[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", "scripts": {}, "answer": "{a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "\n  <p>Find the following indefinite integrals.</p>\n  <p>Input all numbers as fractions or integers and not decimals.</p>\n  ", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "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": ""}, "a1": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}}, "metadata": {"notes": "\n    \t\t<p><strong>3/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Got rid of redundant instructions about inputting constant of integration.</p>\n    \t\t<p>Penalised use of steps in first part, 1 mark. Added message to that effect in first part.</p>\n    \t\t<p>Added message about not inputting decimals in appropriate places.</p>\n    \t\t<p>Changed marks reflecting the use of steps and degree of difficulty in second part.</p>\n    \t\t<p>Improved Advice display.</p>\n    \t\t", "description": "<p>Given $\\displaystyle \\int (ax+b)e^{cx}\\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\\displaystyle \\int (ax+b)^2e^{cx}\\;dx =h(x)e^{cx}+C$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Phil Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/624/"}]}]}], "contributors": [{"name": "Phil Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/624/"}]}