// Numbas version: finer_feedback_settings
{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Integration by Substitution</p>\n<p>rebelmaths</p>\n<p>rebel</p>"}, "timing": {"allowPause": true, "timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}}, "percentPass": 0, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-ordered", "name": "Group", "questions": [{"name": "cormac's copy of Julie's copy of Definite Integration 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "showQuestionGroupNames": false, "parts": [{"prompt": "\n      <p>\\[I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans1-tol", "marks": 1.0, "maxvalue": "ans1+tol"}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n      <p>\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]</p>\n      <p>$I=\\;\\;$[[0]]</p>\n      <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans2-tol", "marks": 1.0, "maxvalue": "ans2+tol"}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n      <p>\\[I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]</p>\n        <p>$I=\\;\\;$[[0]]</p>\n        <p>Input your answer to 3 decimal places.</p>\n      ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans3-tol", "marks": 1.0, "maxvalue": "ans3+tol"}], "marks": 0.0, "type": "gapfill"}], "statement": "<p>Evaluate the following definite integrals.</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n    \t\t                \t\t<p><strong>3/07/1012:</strong></p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>Checked calculations.</p>\n    \t\t                \t\t        <p>Left tolerances in, as easy to make minor errors in calculations.</p>\n    \t\t                \t\t        <p>Improved display in Advice.</p>\n    \t\t                \t\t        <p>Some superscripts e.g. the form a^\\var{b} in latex have to be written as a^{\\var{b}} as not displayed properly (if b has a second digit it slips down). Corrected.</p>\n    \t\t                \t\t        <p><strong>20/07/2012:</strong></p>\n    \t\t                \t\t        <p>Set new tolerace variables, tol=0.01, tol1=0.0001.</p>\n    \t\t                \t\t        <p>Can have expressions in Advice of the form $1\\times E$ where E is an expression. This can be remedied by rewriting - but later as not crucial.</p>\n    \t\t                \t\t        <p>Added description.</p>\n    \t\t                \t\t        <p></p>\n    \t\t                \t\t        <p><strong>25/07/2012:</strong></p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Added tags.</p>\n    \t\t                \t\t        <p>A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>Question appears to be working correctly.</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \t\t        <p>&nbsp;</p>\n    \t\t                \n    \t\t", "description": "<p>Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.&nbsp;</p>"}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "advice": "\n  <p>&nbsp;</p>\n    <p>b)<br/>\\[\\begin{eqnarray*}I&amp;=&amp;\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &amp;=&amp;\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &amp;=&amp; \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &amp;=&amp;\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &amp;=&amp;\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]</p>\n    <p>&nbsp;</p>\n  ", "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "variables": {"d1": {"definition": "random(-9..9)", "name": "d1"}, "b": {"definition": "random(2..5)", "name": "b"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "t": {"definition": "random(1,-1)", "name": "t"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "ans3": {"definition": "precround(tans3,3)", "name": "ans3"}, "tans1": {"definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "a": {"definition": "random(-2..2#0.5 except 0)", "name": "a"}, "b1": {"definition": "random(-1..2#0.5 except 0)", "name": "b1"}, "tans3": {"definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3"}, "w": {"definition": "random(0,1)", "name": "w"}, "ans2": {"definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "tol": {"definition": 0.001, "name": "tol"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}}, "type": "question", "functions": {}, "variable_groups": [], "progress": "ready", "question_groups": [{"questions": [], "name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered"}]}, {"name": "Integration by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Integration by substitution. Hint given on susbtitution</p>\n<p>rebelmaths</p>"}, "statement": "<p>Complete the following indefinite integrals using integration by substition and the letter C for any unknown constants.</p>", "ungrouped_variables": ["c", "a", "f", "d", "b"], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"c": {"name": "c", "definition": "random(5..12)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "f": {"name": "f", "definition": "random(3..5 except d)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"name": "b", "definition": "random(1..10)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"name": "a", "definition": "random(2..6)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "d": {"name": "d", "definition": "random(2..9)", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "tags": ["rebelmaths"], "preamble": {"js": "", "css": ""}, "rulesets": {}, "variable_groups": [], "advice": "<p>Use the substitution given</p>", "parts": [{"checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showpreview": true, "prompt": "<p>$\\int \\cos(\\var{a}x) \\mathrm{dx}$ using the substitution $u = \\var{a}x$.</p>", "variableReplacements": [], "checkingaccuracy": 0.001, "answer": "sin({a}x)/{a} + C", "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "vsetrangepoints": 5}, {"checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showpreview": true, "prompt": "<p>$\\int x(\\var{b}+x^2)^\\var{c}\\mathrm{dx}$ using the substitution $u = \\var{b} + x^2$.</p>", "variableReplacements": [], "checkingaccuracy": 0.001, "answer": "({b}+x^2)^({c}+1)/(2({c}+1))+C", "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "vsetrangepoints": 5}, {"checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showpreview": true, "prompt": "<p>$\\int\\frac{\\mathrm{dt}}{(1-\\var{d}t)^\\var{f}}$ using the substitution $u = 1-\\var{d}t$</p>", "variableReplacements": [], "checkingaccuracy": 0.001, "answer": "1/({d}({f}-1)(1-{d}t)^({f}-1))+C", "expectedvariablenames": [], "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "vsetrangepoints": 5}], "type": "question"}, {"name": "Julie's copy of Leicester: Integration1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "advice": "", "parts": [{"checkingtype": "absdiff", "answer": "{a*c}/{b+c}*x^({b+c}/{c})+C", "vsetrange": [0.0, 1.0], "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "marks": 1.0, "vsetrangepoints": 5.0, "prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[std]{{a}*x^({b}/{c})}$</p>\n        <p>Input $\\displaystyle \\int f(x)\\;dx$ here.</p>\n        <p>&nbsp;</p>\n      ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>"}}, {"checkingtype": "absdiff", "answer": "(1/{b})*({-t[0]}*cos({b}x+{c})+{t[1]}*sin({b}x+{c})+{t[2]}*exp({b}x+{c}))+C", "vsetrange": [0.0, 1.0], "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "marks": 1.0, "vsetrangepoints": 5.0, "prompt": "\n      <p>$f(x)=\\simplify[std]{{t[0]}*sin({b}x+{c})+{t[1]}*cos({b}x+{c})+{t[2]}*exp({b}x+{c})}$</p>\n      <p>Input $\\displaystyle \\int f(x)\\;dx$ here.</p>\n      "}, {"checkingtype": "absdiff", "answer": "{a*c}/{b}*exp({b}/{c}*x)+C", "vsetrange": [0.0, 1.0], "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "marks": 1.0, "vsetrangepoints": 5.0, "prompt": "\n      <p>$f(x)=\\simplify[std]{{a}exp({b}/{c}*x)}$</p>\n      <p>Input $\\displaystyle \\int f(x)\\;dx$ here.</p>\n      ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>"}}, {"checkingtype": "absdiff", "answer": "{a1}/{b1}ln(abs({b1}x+{c1}))+C", "vsetrange": [0.0, 1.0], "type": "jme", "checkingaccuracy": 0.001, "answersimplification": "std", "marks": 1.0, "vsetrangepoints": 5.0, "prompt": "\n      <p>$\\displaystyle f(x)=\\simplify[std]{{a1}/({b1}x+{c1})}$</p>\n        <p>Input $\\displaystyle \\int f(x)\\;dx$ here.</p>\n        <p>&nbsp;</p>\n      ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>"}}], "type": "question", "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "pickQuestions": 0, "name": ""}], "progress": "ready", "tags": [], "metadata": {"notes": "", "description": "<p>Integrating simple functions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "variables": {"c1": {"name": "c1", "definition": "chcp(b1,2)"}, "a": {"name": "a", "definition": "random(-9..9 except [-1,0,1])"}, "t": {"name": "t", "definition": "switch(u=1,[1,0,0],u=2,[0,1,0],[0,0,1])"}, "u": {"name": "u", "definition": "random(1,2,3)"}, "b": {"name": "b", "definition": "chcp(c,2)"}, "a1": {"name": "a1", "definition": "random(-9..9 except[0,a])"}, "c": {"name": "c", "definition": "random(2..9)"}, "b1": {"name": "b1", "definition": "random(2..9)"}}, "functions": {"chcp": {"language": "jme", "parameters": [["a", "number"], ["b", "number"]], "type": "number", "definition": "if(gcd(a,b)=1,b,chcp(a,random(2..9)))"}}, "statement": "\n  <p>Integrate the following functions $f(x)$.</p>\n    <p>Input all numbers as integers or fractions and not as decimals.</p>\n    <p>In all examples do not forget to include the constant of integration $C$.</p>\n  ", "showQuestionGroupNames": false, "variable_groups": []}, {"name": "Integration by substitution 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "metadata": {"description": "<p>Integration by susbtitution, no hint given</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "tags": [], "variables": {"b": {"description": "", "definition": "random(1..8 except a)", "name": "b", "templateType": "anything", "group": "Ungrouped variables"}, "a": {"description": "", "definition": "random(2..6)", "name": "a", "templateType": "anything", "group": "Ungrouped variables"}, "c": {"description": "", "definition": "random(1..9)", "name": "c", "templateType": "anything", "group": "Ungrouped variables"}}, "parts": [{"prompt": "<p>$\\int e^x\\sqrt{1+e^x}\\mathrm{dx}$</p>", "checkingaccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "type": "jme", "showCorrectAnswer": true, "answer": "2(1+e^x)^(3/2)/3+C", "scripts": {}, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": [], "showFeedbackIcon": true}, {"prompt": "<p>$\\int\\frac{\\mathrm{dx}}{\\var{a}x+\\var{b}}$.</p>\n<p>Answer in terms of the natural log, represented by ln( ).</p>", "checkingaccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "type": "jme", "showCorrectAnswer": true, "answer": "1/{a}ln({a}x+{b})+C", "scripts": {}, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": [], "showFeedbackIcon": true}, {"prompt": "<p>$\\int \\frac{x \\mathrm{dx}}{\\var{c}+x^2}$.</p>\n<p>Answer in terms of the natural log, represented by ln( ).</p>", "checkingaccuracy": 0.001, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showpreview": true, "type": "jme", "showCorrectAnswer": true, "answer": "ln({c}+x^2)/2+C", "scripts": {}, "vsetrange": [0, 1], "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": [], "showFeedbackIcon": true}], "variable_groups": [], "ungrouped_variables": ["c", "b", "a"], "advice": "<p>integration by Susbtitution</p>", "statement": "<p>Evaluate the following indefinite integrals using integration by substitution. Use the letter C to represent any unknown constants.</p>", "type": "question"}, {"name": "Julie's copy of Hannah's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "advice": "\n\t<p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$<br/>Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.</p>\n\t  <p>Hence the integral becomes:</p>\n\t  <p>\\[\\begin{eqnarray*} I&amp;=&amp;\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\ &amp;=&amp;\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\ &amp;=&amp; \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C} \\end{eqnarray*}\\]</p>\n\t  <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int g(u)\\;du \\]<br/>And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.</p>\n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n\t\t\t<p>\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]</p>\n\t\t\t      <p>$I=\\;$[[0]]</p>\n\t\t\t      <p>Input numbers in your answer as integers or fractions and not as decimals.</p>\n\t\t\t", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "checkvariablenames": false, "type": "jme"}], "type": "gapfill", "marks": 0.0}], "statement": "\n\t    <p>Find the following integral.</p>\n\t    <p>Input the constant of integration as $C$.</p>\n\t    \n\t  \n\t", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..5)", "name": "a"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "b": {"definition": "s1*random(1..9)", "name": "b"}, "m": {"definition": "random(4..9)", "name": "m"}}, "metadata": {"notes": "\n\t\t    \t\t    \t\t<p><strong>2/08/2012:</strong></p>\n\t\t    \t\t    \t\t    <p>Added tags.</p>\n\t\t    \t\t    \t\t    <p>Added description.</p>\n\t\t    \t\t    \t\t    <p>Checked calculation. OK.</p>\n\t\t    \t\t    \t\t    <p>Added information about Show steps in prompt content area.&nbsp;</p>\n\t\t    \t\t    \t\t    <p>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n\t\t    \t\t    \t\t    <p>Got rid of a redundant ruleset.</p>\n\t\t    \t\t    \t\t    <p>&nbsp;</p>\n\t\t    \t\t    \t\t    <p>&nbsp;</p>\n\t\t    \t\t    \n\t\t    \n\t\t", "description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Julie's copy of Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "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>Corrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.</p>\n\t\t    \t\t<p>Checked calculation. OK.</p>\n\t\t    \t\t<p>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n\t\t    \t\t<p>Message about Show steps included. Also another message about including the constant of integration.</p>\n\t\t    \t\t<p>Changed checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.</p>\n\t\t    \t\t<p>Improved display of Steps by aligning integral signs.</p>\n\t\t    \n\t\t", "description": "<p>Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "pickQuestions": 0, "name": ""}], "parts": [{"type": "gapfill", "steps": [{"marks": 0.0, "type": "information", "prompt": "<p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>"}], "stepspenalty": 0.0, "marks": 0.0, "gaps": [{"checkingtype": "absdiff", "vsetrange": [1.0, 2.0], "type": "jme", "checkingaccuracy": 0.001, "answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "showpreview": true, "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "<p>Input all numbers as integers or fractions and not decimals.</p>"}, "answersimplification": "std", "marks": 2.0, "vsetrangepoints": 5.0, "checkvariablenames": false, "expectedvariablenames": []}], "prompt": "\n  <p>$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$</p>\n<p>$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]</p>\n<p>Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.</p>\n<p>Click on Show steps to get more information. You will not lose any marks by doing so.</p>\n  \n"}], "type": "question", "progress": "ready", "variables": {"s2": {"name": "s2", "definition": "random(1,-1)"}, "a": {"name": "a", "definition": "s1*random(2..5)"}, "s5": {"name": "s5", "definition": "random(1,-1)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}, "c3": {"name": "c3", "definition": "s5*random(2..8)"}, "b1": {"name": "b1", "definition": "s3*random(2..9)"}, "b": {"name": "b", "definition": "s2*random(2..9)"}, "a1": {"name": "a1", "definition": "random(2..5)"}, "a2": {"name": "a2", "definition": "s4*random(3..9)"}, "s4": {"name": "s4", "definition": "random(1,-1)"}, "s3": {"name": "s3", "definition": "random(1,-1)"}}, "tags": ["Calculus", "Steps", "calculus", "constant of integration", "exponential function", "indefinite integration", "integrals", "integrating powers", "integration", "integration of exponential function", "integration of powers", "integration of trigonometric functions", "standard integrals", "steps", "trigonometric functions"], "advice": "\n  <p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>\n  <p>Splitting the integral into three parts and using the above information we have: <br/>\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&amp;=&amp;\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &amp;=&amp;\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]</p>\n  \n", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "variable_groups": [], "statement": "\n  <p>Integrate the following function $f(x)$.</p>\n  <p>&nbsp;<br/>Input the constant of integration as $C$.</p>\n  \n", "showQuestionGroupNames": false, "functions": {}}, {"name": "Julie's copy of Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "statement": "\n\t  \n\t  \n\t  <p>Find the following indefinite integral.</p>\n\t  \n\t  \n\t  \n\t  <p>Input the constant of integration as $C$.</p>\n\t  \n\t  \n\t  \n\t", "progress": "ready", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$</p>", "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>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n\t\t    \t\t<p>Added a Step and message about Show steps included - losing 1 mark if used as it gives the formula for finding the integral. Increased marks to 3 for the question, so that can cope with losing a mark for using Show steps.</p>\n\t\t    \t\t<p>Changed accuracy setting to relative difference of 0.00001 as we have negative powers.</p>\n\t\t    \t\t<p>Checked calculation. OK.</p>\n\t\t    \t\t<p>Added message in prompt &nbsp;about including the constant of integration.</p>\n\t\t    \t\t<p>Noted issue with steps-answer order and the messages/marks generated.</p>\n\t\t    \t\t<p>Changed numerator to the range 2..5.</p>\n\t\t    \t\t<p>Improved display in Advice.</p>\n\t\t    \t\t<p>&nbsp;</p>\n\t\t    \n\t\t"}, "parts": [{"stepspenalty": 1.0, "gaps": [{"answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "vsetrangepoints": 5.0, "checkingtype": "reldiff", "vsetrange": [0.0, 1.0], "marks": 3.0, "type": "jme", "notallowed": {"message": "<p>Input all numbers as integers or fractions and not decimals.</p>", "strings": ["."], "showstrings": false, "partialcredit": 0.0}, "answersimplification": "std", "checkingaccuracy": 0.0001}], "prompt": "\n\t\t\t      <p>$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]</p>\n\t\t\t        <p>Input all numbers as integers or fractions and not decimals. Remember to&nbsp;include the constant of integration $C$.</p>\n\t\t\t        <p>Click on Show steps to get help. You will lose 1 mark by doing so.</p>\n\t\t\t      \n\t\t\t", "marks": 0.0, "type": "gapfill", "steps": [{"marks": 0.0, "type": "information", "prompt": "<p>&nbsp;\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]</p>"}]}], "variables": {"d": {"name": "d", "definition": "random(1..9)"}, "b": {"name": "b", "definition": "random(2..5)"}, "a": {"name": "a", "definition": "random(2..9)"}, "n": {"name": "n", "definition": "random(3..5)"}}, "tags": ["Calculus", "Steps", "calculus", "constant of integration", "indefinite integration", "integrals", "integration", "integration by substitution", "standard integrals", "steps", "substitution"], "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "name": "", "questions": []}], "type": "question", "variable_groups": [], "advice": "\n\t  <p>Let $y = \\simplify[std]{{a}*x+{d}}$. Then,<br/>\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\\]</p>\n\t    <p>Now,<br/>\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\\]</p>\n\t    <p>Rearrange $y = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(y-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{dy} = \\frac{1}{\\var{a}}$.</p>\n\t    <p>$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:</p>\n\t    <p>\\[\\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy = \\simplify[std]{{b}/({n-1}*y^{n-1})} \\cdot \\frac{1}{\\var{a}} + C = \\simplify[std]{(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}.\\]</p>\n\t  \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "showQuestionGroupNames": false}, {"name": "Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "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"}}, "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", "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"}, {"name": "Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "metadata": {"notes": "\n\t\t        \t\t    \t\t<p><strong>2/08/2012:</strong></p>\n\t\t        \t\t    \t\t<p>Added tags.</p>\n\t\t        \t\t    \t\t<p>Added description.</p>\n\t\t        \t\t    \t\t<p>Checked calculation. OK.</p>\n\t\t        \t\t    \t\t<p>Added information about Show steps in prompt content area.&nbsp;</p>\n\t\t        \t\t    \t\t<p>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n\t\t        \t\t    \t\t<p>Got rid of a redundant ruleset.&nbsp;!noLeadingMinus added to std ruleset.</p>\n\t\t        \t\t    \t\t<p>Note that the choice of variables means that the argument of the log answer is always $\\gt 0$ so no need to use abs.</p>\n\t\t        \t\t    \t\t<p>&nbsp;</p>\n\t\t        \t\t    \n\t\t        \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$</p>"}, "showQuestionGroupNames": false, "advice": "\n\t<p>This exercise is best solved by using substitution.</p>\n\t  <p>Note that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$</p>\n\t  <p>So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$</p>\n\t  <p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n\t  <p>Hence the integral becomes:</p>\n\t  <p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int\\;\\frac{du}{u}\\\\ &amp;=&amp;\\ln(|u|)+C\\\\ &amp;=&amp; \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]<br/><br/></p>\n\t  <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]</p>\n\t", "functions": {}, "progress": "ready", "tags": ["Steps", "calculus", "indefinite integration", "integration", "integration by substitution", "substitution"], "variables": {"test": {"definition": "4*a*c-b^2", "name": "test"}, "f": {"definition": "-a*(1+b1)^2", "name": "f"}, "b": {"definition": "2*a+b1", "name": "b"}, "a": {"definition": "random(1..5)", "name": "a"}, "c": {"definition": "a*b1^2+c1", "name": "c"}, "c1": {"definition": "max(-10,f+1)+random(1..5)", "name": "c1"}, "b1": {"definition": "s1*random(1..5)", "name": "b1"}, "s1": {"definition": "random(1,-1)", "name": "s1"}}, "parts": [{"prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]</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>Input all numbers as integers or fractions not as decimals.</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        \n\t\t\t", "stepspenalty": 1.0, "marks": 0.0, "steps": [{"prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>", "marks": 0.0, "type": "information"}], "type": "gapfill", "gaps": [{"type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "answersimplification": "std", "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+C", "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}, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "marks": 3.0, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": []}]}], "question_groups": [{"pickQuestions": 0, "pickingStrategy": "all-ordered", "name": "", "questions": []}], "type": "question", "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.</p>\n\t    <p>&nbsp;</p>\n\t  \n\t    \n\t"}, {"name": "cormac's copy of Julie's copy of Indefinite integral by substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "statement": "\n\t  <p>Find the following integral.</p>\n\t  <p>Input the constant of integration as $C$.</p>\n\t  \n\t", "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "name": "", "questions": []}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "progress": "ready", "variable_groups": [], "parts": [{"stepspenalty": 1.0, "marks": 0.0, "prompt": "\n\t\t\t      <p>\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]</p>\n\t\t\t      <p>$I=\\;$[[0]]</p>\n\t\t\t      <p>Input numbers in your answer as integers or fractions and not as decimals.</p>\n\t\t\t      <p>Click on Show steps to get further help. You will lose 1 mark if you do so.</p>\n\t\t\t      \n\t\t\t", "type": "gapfill", "steps": [{"type": "information", "marks": 0.0, "prompt": "<p>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$</p>"}], "gaps": [{"answersimplification": "std", "checkingaccuracy": 0.001, "type": "jme", "checkingtype": "absdiff", "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C", "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "notallowed": {"showstrings": false, "partialcredit": 0.0, "message": "<p>Input all numbers as integers or fractions and not as decimals.</p>", "strings": ["."]}, "marks": 3.0}]}], "variables": {"a": {"name": "a", "definition": "random(1..5)"}, "b": {"name": "b", "definition": "s1*random(1..9)"}, "m": {"name": "m", "definition": "random(4..9)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}}, "advice": "\n\t    \n\t    \n\t    <p>This exercise is best solved by using substitution. <br/>Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$<br/>Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.</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/{2*a})u^{m},u)}\\\\\n\t    \n\t    &#38;=&#38;\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n\t    \n\t    &#38;=&#38; \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n\t    \n\t    \\end{eqnarray*}\\]</p>\n\t    \n\t    \n\t    \n\t    <p><b>A Useful Result</b><br/>This example can be generalised. <br/>Suppose \\[I = \\int\\; f&#39;(x)g(f(x))\\;dx\\]<br/>The using the substitution $u=f(x)$ we find that $du=f&#39;(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int g(u)\\;du \\]<br/>And if we can find this simpler integral in terms of $u$ we can replace $u$ by $f(x)$ and get the result in terms of $x$.</p>\n\t    \n\t    \n\t  \n\t", "showQuestionGroupNames": false, "metadata": {"description": "<p>Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$</p>", "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>Added decimal point as forbidden string and included message in prompt about not entering decimals.</p>\n\t\t    \t\t    <p>Got rid of a redundant ruleset.</p>\n\t\t    \t\t    <p>&nbsp;</p>\n\t\t    \t\t    <p>&nbsp;</p>\n\t\t    \n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "functions": {}}, {"name": "cormac's copy of Definite Integrals 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "parts": [{"vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showFeedbackIcon": true, "showpreview": true, "answer": "2/pi", "prompt": "<p>$\\int_0^1\\cos(\\frac{\\pi t}{2})\\mathrm{dt}$.</p>\n<p>To write $\\pi$ in your answer simply write pi.</p>", "checkingtype": "absdiff"}, {"vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "marks": 1, "variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showFeedbackIcon": true, "showpreview": true, "answer": "ln(e+1)", "prompt": "<p>$\\int_0^1\\frac{e^z+1}{e^z+z}\\mathrm{dz}$.</p>\n<p>Express your answer using the natural log, ln().</p>\n<p>Hint: make a substition using the lower line.</p>", "checkingtype": "absdiff"}], "tags": [], "metadata": {"description": "<p>Definite Integrals</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "ungrouped_variables": [], "rulesets": {}, "variables": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "advice": "<p>Definite Integrals</p>", "functions": {}, "statement": "<p>Find the following definite integrals</p>", "type": "question"}]}], "navigation": {"preventleave": true, "browse": true, "showresultspage": "oncompletion", "reverse": true, "allowregen": true, "showfrontpage": true, "onleave": {"message": "", "action": "none"}}, "duration": 0, "name": "Tien Chern's copy of  Integration by Substitution Exam", "showstudentname": true, "feedback": {"allowrevealanswer": true, "intro": "", "advicethreshold": 0, "showtotalmark": true, "showactualmark": true, "feedbackmessages": [], "showanswerstate": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "showQuestionGroupNames": false, "type": "exam", "contributors": [{"name": "Tien Chern Chia", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1601/"}], "extensions": [], "custom_part_types": [], "resources": []}