// Numbas version: exam_results_page_options
{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>4 questions on integrating by parts.</p>"}, "timing": {"timeout": {"message": "", "action": "none"}, "allowPause": true, "timedwarning": {"message": "", "action": "none"}}, "navigation": {"onleave": {"message": "", "action": "none"}, "preventleave": false, "showfrontpage": false, "browse": true, "showresultspage": "oncompletion", "allowregen": true, "reverse": true}, "question_groups": [{"pickQuestions": 1, "name": "Group", "pickingStrategy": "all-ordered", "questions": [{"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "exponential function", "integrals", "integration", "integration by parts", "integration of exponential function", "steps"], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <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 lose 1 mark if you do.</p>\n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}/{c})*x+{c*b-a}/{c^2}", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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          ", "marks": 0}], "type": "gapfill"}, {"prompt": "\n      <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>\n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "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", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "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  ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "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"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1"], "tags": ["Calculus", "Steps", "calculus", "constant of integration", "indefinite integration", "integrals", "integrating", "integrating trigonometric functions", "integration by parts", "steps", "twice"], "preamble": {"css": "", "js": ""}, "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 $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{cos({c}*x+{d})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx} = \\simplify[std]{{a}}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{c})*sin({c}*x+{d})}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*cos({c}*x+{d})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(({a}*x+{b})/{c})*sin({c}*x+{d}) - ({a}/{c})*Int(sin({c}*x+{d}),x)} \\\\ &amp;=&amp; \\simplify[std]{(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}) + C} \\end{eqnarray} \\]</p>\n<h4>b)</h4>\n<p>For this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle&nbsp;\\frac{dv}{dx} = \\simplify[std]{sin({c}*x+{d})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx}=\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle&nbsp;v = \\simplify[std]{-(1/{c})*cos({c}*x+{d})}$.</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*cos({c}*x+{d}) + (1/{c})*Int({2*a}*({a}x+{b})*cos({c}*x+{d}),x)} \\\\ &amp;=&amp; \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*Int(({a}x+{b})*cos({c}*x+{d}),x)}\\dots (*) \\end{eqnarray*}\\]</p>\n<p>But in Part a we have aready worked out $\\displaystyle&nbsp;\\simplify[std]{Int(({a}x+{b})*cos({c}*x+{d}),x)=(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d})}$&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*cos({c}*x+{d}) +({2*a}/{c})*((({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}))+C}\\\\ &amp;=&amp; \\simplify[std]{-(({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C} \\end{eqnarray*}\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n        <p>$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx $</p>\n        <p>$I=\\;$[[0]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n        <p>You can get help by clicking on Show steps. You will lose 1 mark if you do.</p>\n        \n      ", "marks": 0, "gaps": [{"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({a}*x+{b})/{c}*sin({c}*x+{d})+{a}/{c^2}*cos({c}*x+{d})+C", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "prompt": "\n          \n          \n          <p>The formula for integrating by parts is<br/>\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]</p>\n          \n          \n            \n          ", "marks": 0}], "type": "gapfill"}, {"prompt": "\n        <p>Use the result from the first part to find:</p>\n        <p>$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*sin({c}*x+{d})} dx $</p>\n        <p>$I=\\;$[[0]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n        \n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "(-({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C", "marks": 4, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "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    <p>Input the constant of integration as $C$.</p>\n    \n  ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "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": ""}, "d": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "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": ""}}, "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>Got rid of redundant ruleset, added !noLeadingMinus to std ruleset as we need to keep the standard order for integrating by parts.</p>\n    \t\t    \t\t<p>Checked calculation. OK.</p>\n    \t\t    \t\t<p>Penalised use of steps in first part, 1 mark. Added message to that effect in first part.</p>\n    \t\t    \t\t<p>Added message about not inputting decimals in appropriate places.</p>\n    \t\t    \t\t<p>Changed marks reflecting the use of steps and degree of difficulty in second part.</p>\n    \t\t    \t\t<p>Improved Advice display.</p>\n    \t\t    \n    \t\t", "description": "<p>Find $\\displaystyle \\int (ax+b)\\cos(cx+d)\\; dx $ and hence find $\\displaystyle \\int (ax+b)^2\\sin(cx+d)\\; dx $&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "constant of integration", "integrals", "integrating trigonometric functions", "integration", "integration by parts", "steps"], "preamble": {"css": "", "js": ""}, "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 $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \\int \\left( \\simplify[std]{(-{a}/{b})*cos({b}x+{c})}\\right) dx \\\\ &amp;=&amp; \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \\end{eqnarray} \\]</p>\n<h4>b)</h4>\n<p>For this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.</p>\n<p>So $\\displaystyle&nbsp;\\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle&nbsp;v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.</p>\n<p>Hence,<br/>\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx &amp;=&amp; uv - \\int v \\frac{du}{dx} dx \\\\ &amp;=&amp; \\simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \\int \\left( \\simplify[std]{({a}/{b})*sin({b}x+{c})}\\right) dx \\\\ &amp;=&amp; \\simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \\end{eqnarray} \\]</p>\n<h4>c)</h4>\n<p>Using the results from Parts a and b, we have \\[\\begin{eqnarray*}I &amp;=&amp; \\int \\simplify[std]{{a1}x*sin({b}x+{c})} dx + \\int \\simplify[std]{{a2}x*cos({b}x+{c})} dx\\\\ &amp;=&amp; \\simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\\\ &amp;=&amp;\\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\\\ &amp;=&amp;\\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \\end{eqnarray*}\\]<br/>Hence <br/>$\\displaystyle&nbsp;\\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$</p>\n<p>$\\displaystyle&nbsp;\\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "\n      <p>$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n        <p>Input the constant of integration as $C$.</p>\n        <p>You can get help by clicking on Show steps. You will lose 1 mark if you do.</p>\n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({a}/{b^2})sin({b}x+{c}) - ({a}/{b})*x*cos({b}x+{c}) + C", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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          ", "marks": 0}], "type": "gapfill"}, {"prompt": "\n      <p>$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n        <p>Input the constant of integration as $C$.</p>\n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "type": "gapfill"}, {"prompt": "\n      <p>Using the first two parts find:<br/>$\\displaystyle I=\\int \\simplify[std]{{a1}x*sin({b}x+{c})+{a2}x*cos({b}x+{c})} dx $<br/>You are given that \\[I=\\simplify[std]{f(x)*sin({b}x+{c})+g(x)*cos({b}x+{c})+C}\\]<br/>where $f(x)$ and $g(x)$ are polynomials of degree 1. You have to find $f(x)$ and $g(x)$.</p>\n        <p>$f(x)=\\;$[[0]]&nbsp;$\\;\\;\\;\\;\\;g(x)=\\;$[[1]]</p>\n        <p>Input all numbers as fractions or integers and not decimals.</p>\n      ", "marks": 0, "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}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{a2}/{b}*x+{a1}/{b^2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"notallowed": {"message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{-a1}/{b}*x+{a2}/{b^2}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "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    <p>Input the constant of integration as $C$ where needed.</p>\n  ", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "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": ""}, "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>Corrected error in second question answer, + changed to -. Also solution to second gap in third part. Advice changed accordingly.</p>\n    \t\t<p>Checked calculations after corrections. OK.</p>\n    \t\t<p>Penalised use of steps in first part, 1 mark. Added message to that effect.</p>\n    \t\t<p>Changed marks to allow for steps penalty.</p>\n    \t\t<p>Improved Advice display.</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx&nbsp;$ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}], "functions": {}, "ungrouped_variables": ["s3", "c", "b", "m"], "tags": ["Calculus", "Steps", "algebraic manipulation", "calculus", "indefinite integration", "integrals", "integration", "integration by parts", "steps"], "preamble": {"css": "", "js": ""}, "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": 1, "prompt": "\n        <p>$I=\\displaystyle \\int \\simplify[std]{x*({b}x+{c})^{m}} dx $<br/>You are given that \\[I=\\simplify[std]{({b}x+{c})^{m+1}/{b^2*(m+1)*(m+2)}*g(x)+C}\\]<br/>For a polynomial $g(x)$.&nbsp;You have to find $g(x)$.</p>\n        <p>$g(x)=\\;$[[0]]</p>\n        <p>You can get help by clicking on Show steps. You will lose 1 mark if you do.</p>\n        \n      ", "marks": 0, "gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "{b*(m+1)}*x-{c}", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "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. \\]<br/>Also you need to know that for $n \\neq -1$:<br/>\\[ \\int (ax+b)^n dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]</p>\n          \n          \n            \n          ", "marks": 0}], "type": "gapfill"}], "statement": "<p>Find the following indefinite integral.</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "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"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "duration": 0, "percentPass": "0", "name": "Integration by parts", "feedback": {"showactualmark": true, "advicethreshold": 0, "allowrevealanswer": true, "intro": "", "showtotalmark": true, "showanswerstate": true, "feedbackmessages": []}, "showstudentname": true, "showQuestionGroupNames": false, "type": "exam", "contributors": [{"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}, {"name": "Jack Dunham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2484/"}], "extensions": [], "custom_part_types": [], "resources": []}