Integration by Substitution
\nrebelmaths
\nrebel
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Definite Integration 2", "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": {}, "tags": ["Calculus", "calculus", "definite integration", "integration", "integration by parts", "integration by parts twice"], "advice": "\n\n
b)
\\[\\begin{eqnarray*}I&=&\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &=&\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &=& \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &=&\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &=&\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]
\n ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n
\\[I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 3 decimal places.
\n ", "gaps": [{"minvalue": "ans1-tol", "type": "numberentry", "maxvalue": "ans1+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 3 decimal places.
\n ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n\\[I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 3 decimal places.
\n ", "gaps": [{"minvalue": "ans3-tol", "type": "numberentry", "maxvalue": "ans3+tol", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "statement": "Evaluate the following definite integrals.
", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-2..2#0.5 except 0)", "name": "a"}, "m2": {"definition": "random(1..9)", "name": "m2"}, "b": {"definition": "random(2..5)", "name": "b"}, "w": {"definition": "random(0,1)", "name": "w"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "ans2": {"definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2"}, "ans3": {"definition": "precround(tans3,3)", "name": "ans3"}, "b2": {"definition": "random(1..20)", "name": "b2"}, "b1": {"definition": "random(-1..2#0.5 except 0)", "name": "b1"}, "tol": {"definition": 0.001, "name": "tol"}, "t": {"definition": "random(1,-1)", "name": "t"}, "m3": {"definition": "random(2..9)", "name": "m3"}, "ans1": {"definition": "precround(tans1,3)", "name": "ans1"}, "c1": {"definition": "t*random(1..9)", "name": "c1"}, "tans1": {"definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1"}, "tol1": {"definition": 0.0001, "name": "tol1"}, "tans3": {"definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3"}, "d1": {"definition": "random(-9..9)", "name": "d1"}}, "metadata": {"notes": "\n \t\t \t\t3/07/1012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tChecked calculations.
\n \t\t \t\tLeft tolerances in, as easy to make minor errors in calculations.
\n \t\t \t\tImproved display in Advice.
\n \t\t \t\tSome 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.
\n \t\t \t\t20/07/2012:
\n \t\t \t\tSet new tolerace variables, tol=0.01, tol1=0.0001.
\n \t\t \t\tCan 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.
\n \t\t \t\tAdded description.
\n \t\t \t\t \n \t\t \t\t25/07/2012:
\n \t\t \t\t\n \t\t \t\t
Added tags.
\n \t\t \t\tA lot of work in this question - Perhaps it would be more managable broken down into two separate questions?
\n \t\t \t\t\n \t\t \t\t
Question appears to be working correctly.
\n \t\t \t\t\n \t\t \t\t
\n \t\t \n \t\t", "description": "
Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"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": "Integration by substitution. Hint given on susbtitution
\nrebelmaths
"}, "statement": "Complete the following indefinite integrals using integration by substition and the letter C for any unknown constants.
", "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": "Use the substitution given
", "parts": [{"checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "showpreview": true, "prompt": "$\\int \\cos(\\var{a}x) \\mathrm{dx}$ using the substitution $u = \\var{a}x$.
", "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": "$\\int x(\\var{b}+x^2)^\\var{c}\\mathrm{dx}$ using the substitution $u = \\var{b} + x^2$.
", "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": "$\\int\\frac{\\mathrm{dt}}{(1-\\var{d}t)^\\var{f}}$ using the substitution $u = 1-\\var{d}t$
", "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": "Leicester: Integration1", "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": {"chcp": {"definition": "if(gcd(a,b)=1,b,chcp(a,random(2..9)))", "type": "number", "language": "jme", "parameters": [["a", "number"], ["b", "number"]]}}, "tags": [], "advice": "", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers"]}, "parts": [{"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n$\\displaystyle f(x)=\\simplify[std]{{a}*x^({b}/{c})}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a*c}/{b+c}*x^({b+c}/{c})+C", "type": "jme"}, {"prompt": "\n
$f(x)=\\simplify[std]{{t[0]}*sin({b}x+{c})+{t[1]}*cos({b}x+{c})+{t[2]}*exp({b}x+{c})}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "(1/{b})*({-t[0]}*cos({b}x+{c})+{t[1]}*sin({b}x+{c})+{t[2]}*exp({b}x+{c}))+C", "type": "jme"}, {"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n$f(x)=\\simplify[std]{{a}exp({b}/{c}*x)}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a*c}/{b}*exp({b}/{c}*x)+C", "type": "jme"}, {"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n$\\displaystyle f(x)=\\simplify[std]{{a1}/({b1}x+{c1})}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{a1}/{b1}ln(abs({b1}x+{c1}))+C", "type": "jme"}], "statement": "\n
Integrate the following functions $f(x)$.
\nInput all numbers as integers or fractions and not as decimals.
\nIn all examples do not forget to include the constant of integration $C$.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(-9..9 except [-1,0,1])", "name": "a"}, "c": {"definition": "random(2..9)", "name": "c"}, "b": {"definition": "chcp(c,2)", "name": "b"}, "a1": {"definition": "random(-9..9 except[0,a])", "name": "a1"}, "u": {"definition": "random(1,2,3)", "name": "u"}, "t": {"definition": "switch(u=1,[1,0,0],u=2,[0,1,0],[0,0,1])", "name": "t"}, "c1": {"definition": "chcp(b1,2)", "name": "c1"}, "b1": {"definition": "random(2..9)", "name": "b1"}}, "metadata": {"notes": "", "description": "Integrating simple functions.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"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": "Integration by susbtitution, no hint given
", "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": "$\\int e^x\\sqrt{1+e^x}\\mathrm{dx}$
", "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": "$\\int\\frac{\\mathrm{dx}}{\\var{a}x+\\var{b}}$.
\nAnswer in terms of the natural log, represented by ln( ).
", "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": "$\\int \\frac{x \\mathrm{dx}}{\\var{c}+x^2}$.
\nAnswer in terms of the natural log, represented by ln( ).
", "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": "integration by Susbtitution
", "statement": "Evaluate the following indefinite integrals using integration by substitution. Use the letter C to represent any unknown constants.
", "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\tThis exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$
Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.
Hence the integral becomes:
\n\t\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\ &=&\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\ &=& \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C} \\end{eqnarray*}\\]
\n\tA Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int g(u)\\;du \\]
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 $x$.
\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput numbers in your answer as integers or fractions and not as decimals.
\n\t\t\t", "gaps": [{"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "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\tFind the following integral.
\n\tInput the constant of integration as $C$.
\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\t2/08/2012:
\n\t\t \t\t \t\tAdded tags.
\n\t\t \t\t \t\tAdded description.
\n\t\t \t\t \t\tChecked calculation. OK.
\n\t\t \t\t \t\tAdded information about Show steps in prompt content area.
\n\t\t \t\t \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n\t\t \t\t \t\tGot rid of a redundant ruleset.
\n\t\t \t\t \t\t\n\t\t \t\t \t\t
\n\t\t \t\t \n\t\t \n\t\t", "description": "
Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
", "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\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tCorrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\t \t\tMessage about Show steps included. Also another message about including the constant of integration.
\n\t\t \t\tChanged checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.
\n\t\t \t\tImproved display of Steps by aligning integral signs.
\n\t\t \n\t\t", "description": "Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.
", "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": "Note that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
"}], "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": "Input all numbers as integers or fractions and not decimals.
"}, "answersimplification": "std", "marks": 2.0, "vsetrangepoints": 5.0, "checkvariablenames": false, "expectedvariablenames": []}], "prompt": "\n$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\nClick on Show steps to get more information. You will not lose any marks by doing so.
\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": "\nNote that \\[\\begin{eqnarray*} &\\int& \\;x^n\\;dx&=&\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &\\int& \\;\\sin(ax)\\;dx &=& -\\frac{1}{a}\\cos(ax)+C\\\\ &\\int& \\;e^{ax}\\;dx &=& \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]
\nSplitting the integral into three parts and using the above information we have:
\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&=&\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &=&\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]
Integrate the following function $f(x)$.
\n
Input the constant of integration as $C$.
Find the following indefinite integral.
\n\t \n\t \n\t \n\tInput the constant of integration as $C$.
\n\t \n\t \n\t \n\t", "progress": "ready", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$
", "notes": "\n\t\t \t\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n\t\t \t\tAdded 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.
\n\t\t \t\tChanged accuracy setting to relative difference of 0.00001 as we have negative powers.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tAdded message in prompt about including the constant of integration.
\n\t\t \t\tNoted issue with steps-answer order and the messages/marks generated.
\n\t\t \t\tChanged numerator to the range 2..5.
\n\t\t \t\tImproved display in Advice.
\n\t\t \t\t\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": "
Input all numbers as integers or fractions and not decimals.
", "strings": ["."], "showstrings": false, "partialcredit": 0.0}, "answersimplification": "std", "checkingaccuracy": 0.0001}], "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.
\n\t\t\tClick on Show steps to get help. You will lose 1 mark by doing so.
\n\t\t\t \n\t\t\t", "marks": 0.0, "type": "gapfill", "steps": [{"marks": 0.0, "type": "information", "prompt": "\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]
"}]}], "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\tLet $y = \\simplify[std]{{a}*x+{d}}$. Then,
\\[\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\\]
Now,
\\[\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\\]
Rearrange $y = \\simplify[std]{{a}x+{d}}$ to get $\\displaystyle x = \\simplify[std]{(y-{b})/{a}}$, and hence $\\displaystyle\\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\n\t$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:
\n\t\\[\\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}.\\]
\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\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tAdded information about Show steps in prompt content area.
\n\t\t \t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$
"}, "showQuestionGroupNames": false, "advice": "\n\t \n\t \n\tThis exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a}+{b}cos(x)}$ then $du=\\simplify[std]{({-b}*sin(x))*dx }$
Hence we can replace $\\sin(x)\\;dx$ by $\\frac{1}{\\var{-b}}\\;du$.
Hence the integral becomes:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{-b})u^{m},u)}\\\\\n\t \n\t &=&\\simplify[std]{(1/{-b})u^{m+1}/{m+1}+C}\\\\\n\t \n\t &=& \\simplify[std]{({a}+{b}*cos(x))^{m+1}/{-b*(m+1)}+C}\n\t \n\t \\end{eqnarray*}\\]
\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\\[I=\\simplify[std]{Int( sin(x)*({a} + {b}*cos(x))^{m},x)}\\]
\n\t\t\tInput all numbers as integers or fractions.
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tClick on Show steps if you need help. You will lose 1 mark if you do so.
\n\t\t\t \n\t\t\t", "stepspenalty": 1.0, "marks": 0.0, "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a}+{b}cos(x)}$
", "marks": 0.0, "type": "information"}], "type": "gapfill", "gaps": [{"notallowed": {"strings": ["."], "partialcredit": 0.0, "message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions.
", "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\tFind the following integral.
\n\tInput the constant of integration as $C$.
\n\tInput all numbers as integers or fractions not as decimals.
\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\t2/08/2012:
\n\t\t \t\t \t\tAdded tags.
\n\t\t \t\t \t\tAdded description.
\n\t\t \t\t \t\tChecked calculation. OK.
\n\t\t \t\t \t\tAdded information about Show steps in prompt content area.
\n\t\t \t\t \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n\t\t \t\t \t\tGot rid of a redundant ruleset. !noLeadingMinus added to std ruleset.
\n\t\t \t\t \t\tNote that the choice of variables means that the argument of the log answer is always $\\gt 0$ so no need to use abs.
\n\t\t \t\t \t\t\n\t\t \t\t \n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$
"}, "showQuestionGroupNames": false, "advice": "\n\tThis exercise is best solved by using substitution.
\n\tNote 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 $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$
\n\tSo if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$
\n\tHence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$
\n\tHence the integral becomes:
\n\t\\[\\begin{eqnarray*} I&=&\\int\\;\\frac{du}{u}\\\\ &=&\\ln(|u|)+C\\\\ &=& \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]
A Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]
\\[I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tInput all numbers as integers or fractions not as decimals.
\n\t\t\tClick on Show steps if you need help. You will lose 1 mark if you do so.
\n\t\t\t \n\t\t\t \n\t\t\t", "stepspenalty": 1.0, "marks": 0.0, "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$
", "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": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "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\tFind the following integral.
\n\tInput the constant of integration as $C$.
\n\tInput all numbers as integers or fractions.
\n\t\n\t \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/"}], "parts": [{"prompt": "\n\t\t\t
\\[I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\\]
\n\t\t\t$I=\\;$[[0]]
\n\t\t\tInput numbers in your answer as integers or fractions and not as decimals.
\n\t\t\tClick on Show steps to get further help. You will lose 1 mark if you do so.
\n\t\t\t \n\t\t\t", "marks": 0.0, "steps": [{"prompt": "Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$
", "type": "information", "marks": 0.0}], "stepspenalty": 1.0, "gaps": [{"checkingtype": "absdiff", "answersimplification": "std", "checkingaccuracy": 0.001, "notallowed": {"strings": ["."], "message": "Input all numbers as integers or fractions and not as decimals.
", "partialcredit": 0.0, "showstrings": false}, "type": "jme", "vsetrange": [0.0, 1.0], "marks": 3.0, "vsetrangepoints": 5.0, "answer": "({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C"}], "type": "gapfill"}], "progress": "ready", "question_groups": [{"pickQuestions": 0, "name": "", "questions": [], "pickingStrategy": "all-ordered"}], "metadata": {"description": "Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$
", "notes": "\n\t\t \t\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tAdded information about Show steps in prompt content area.
\n\t\t \t\tAdded decimal point as forbidden string and included message in prompt about not entering decimals.
\n\t\t \t\tGot rid of a redundant ruleset.
\n\t\t \t\t\n\t\t \t\t
\n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "variables": {"a": {"name": "a", "definition": "random(1..5)"}, "m": {"name": "m", "definition": "random(4..9)"}, "s1": {"name": "s1", "definition": "random(1,-1)"}, "b": {"name": "b", "definition": "s1*random(1..9)"}}, "tags": ["Calculus", "Steps", "calculus", "indefinite integration", "integrals", "integration", "integration by substitution", "steps", "substitution"], "statement": "\n\t
Find the following integral.
\n\tInput the constant of integration as $C$.
\n\t \n\t", "advice": "\n\t \n\t \n\tThis exercise is best solved by using substitution.
Note that if we let $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$ then $du=\\simplify[std]{({2*a} * x)*dx }$
Hence we can replace $xdx$ by $\\frac{1}{\\var{2*a}}du$.
Hence the integral becomes:
\n\t \n\t \n\t \n\t\\[\\begin{eqnarray*} I&=&\\simplify[std]{Int((1/{2*a})u^{m},u)}\\\\\n\t \n\t &=&\\simplify[std]{(1/{2*a})u^{m+1}/{m+1}+C}\\\\\n\t \n\t &=& \\simplify[std]{({a} * (x ^ 2) + {b})^{m+1}/{2*a*(m+1)}+C}\n\t \n\t \\end{eqnarray*}\\]
\n\t \n\t \n\t \n\tA Useful Result
This example can be generalised.
Suppose \\[I = \\int\\; f'(x)g(f(x))\\;dx\\]
The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:
\\[I = \\int g(u)\\;du \\]
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 $x$.
Let $y = \\simplify[std]{{a}*x+{d}}$.
\n\tThen $\\displaystyle x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ and so we have the numerator $\\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$:
\n\t$\\displaystyle \\simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so
\n\t\\[\\displaystyle \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} = \\simplify[std]{({m}*y+{r})/(y^{n})={m}/y^{n-1}+{r}/y^{n}}\\]
\n\tNow,
\\[\\int \\simplify[std]{({b}x+{c})/({a}*x+{d})^{n}} dx = \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}} \\right)\\frac{dx}{dy} dy \\]
Since $\\displaystyle x = \\simplify[std]{(y-{d})/{a}}$ then $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\n\tWe can now calculate the desired integral:
\n\t\\[ \\begin{eqnarray*} \\int \\left(\\simplify[std]{{m}/y^{n-1}+{r}/y^{n}}\\right) \\frac{dx}{dy} dy &=&\\frac{1}{\\var{a}}\\left(\\int \\simplify[std]{{m}/y^{n-1}}\\;dy+\\int \\simplify[std]{{r}/y^{n}}\\;dy \\right)\\\\ &=&\\frac{1}{\\var{a}}\\left(\\simplify[std]{{-m}/({n-2}*y^{n-2})+ {-r}/({n-1}*y^{n-1})}\\right) + C \\\\ &=& \\simplify[std]{(-{m})/({a*(n-2)}*({a}*x+{d})^{n-2})+(-{r})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{a*(n-2)})*({a}x+{d})-{r}/({a*(n-1)}))}\\\\ &=&\\simplify[std]{1/({a}x+{d})^{n-1}*(({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a})} \\end{eqnarray*} \\]
Hence \\[g(x)=\\simplify[std]{({-m}/{(n-2)})*x+{-m*d*(n-1)-r*(n-2)}/{(n-2)*(n-1)*a}}\\]
$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\n\t\t\tYou are given that \\[I=\\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C\\] for a polynomial $g(x)$. You have to find $g(x)$.
\n\t\t\t$g(x)=\\;$[[0]]
\n\t\t\tRemember to input all numbers as integers or fractions.
\n\t\t\tClick on Show steps to get help if you need it. You will lose 1 mark by doing so.
\n\t\t\t \n\t\t\t", "gaps": [{"notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({-m}/{n-2})*x-{m*d*(n-1)+r*(n-2)}/{(n-2)*(n-1)*a}", "type": "jme"}], "steps": [{"prompt": "One way to do this is by substitution, for example $y = \\simplify[std]{{a}*x+{d}}$.
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n\tFind the following indefinite integral.
\n\tInput all numbers as integers or fractions, not as decimals.
\n\tInput the constant of integration as $C$.
\n\t \n\t", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..5)", "name": "a"}, "c": {"definition": "m*d+r", "name": "c"}, "b": {"definition": "m*a", "name": "b"}, "d": {"definition": "random(1..5)", "name": "d"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "m": {"definition": "s1*random(1..4)", "name": "m"}, "n": {"definition": "random(3..5)", "name": "n"}, "r": {"definition": "s2*random(1..5)", "name": "r"}}, "metadata": {"notes": "\n\t\t \t\t2/08/2012:
\n\t\t \t\tAdded tags.
\n\t\t \t\tAdded description.
\n\t\t \t\tAdded 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.
\n\t\t \t\tChecked calculation. OK.
\n\t\t \t\tImproved display in Advice.
\n\t\t \n\t\t", "description": "Find the polynomial $g(x)$ such that $\\displaystyle \\int \\frac{ax+b}{(cx+d)^{n}} dx=\\frac{g(x)}{(cx+d)^{n-1}}+C$.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Leicester: Integration 2", "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": {}, "tags": ["integration substitution"], "advice": "", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n$\\displaystyle f(x)=\\simplify[std]{({u}*x+{1-u}*(cos({c}x+{d})))*cos({u}*({a}x^2+{b})+{1-u}*(sin({c}x+{d})))}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{u}/{2*a}*sin({a}x^2+{b})+{1-u}/{c}*sin(sin({c}x+{d}))+C", "type": "jme"}, {"notallowed": {"message": "Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n\n
$\\displaystyle f(x)=\\simplify[std]{({v}*x^2+{1-v}*(sin({c1}x+{d1})))*exp({v}*({a1}x^3)+{1-v}*(cos({c1}x+{d1})))}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "(1/{3*a1}*{v}-{1}/{c1}*{1-v})*exp({v}*({a1}x^3)+{1-v}*(cos({c1}x+{d1})))+C", "type": "jme"}, {"notallowed": {"message": "
Input all numbers as integers or fractions and not as decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "prompt": "\n$\\displaystyle f(x)=\\simplify[std]{x^2*({1-u}*({a}+x^3)^({n}/2)+{u}*tan({a1}x^3+{d1}))}$
\nInput $\\displaystyle \\int f(x)\\;dx$ here.
\n ", "checkingaccuracy": 0.001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 1.0, "answer": "{1-u}*{2}/{3*(n+2)}*({a}+x^3)^({n+2}/2)+{u}/{3*a1}*ln(abs(sec({a1}x^3+{d1})))+C", "type": "jme"}], "statement": "\nIntegrate the following functions $f(x)$.
\nInput all numbers as integers or fractions and not as decimals.
\nMake sure you include the constant of integration $C$.
\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..8)", "name": "a"}, "c": {"definition": "random(2..9)", "name": "c"}, "b": {"definition": "random(-9..9 except 0)", "name": "b"}, "d": {"definition": "random(-9..9 except 0)", "name": "d"}, "n": {"definition": "random(3..11#2)", "name": "n"}, "a1": {"definition": "random(2..9)", "name": "a1"}, "u": {"definition": "random(0,1)", "name": "u"}, "v": {"definition": "random(0,1)", "name": "v"}, "c1": {"definition": "random(2..9)", "name": "c1"}, "d1": {"definition": "random(2..9)", "name": "d1"}}, "metadata": {"notes": "Note that we insist that $\\int \\frac{1}{x} \\;dx=\\ln(|u|)+C$ and $\\int \\tan(x) \\;dx=\\ln(|\\sec(u)|)+C$, so that users must input the absolute value as shown. This is crucial in the last part if the integral $\\int x^2 \\tan(x^3+a) \\;dx$ is to be evaluated.
", "description": "Integration using subsitution
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Definite Integrals 3", "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": "Definite Integrals
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variables": {}, "tags": [], "statement": "Find the following definite integrals
", "preamble": {"js": "", "css": ""}, "rulesets": {}, "variable_groups": [], "advice": "Definite Integrals
", "parts": [{"checkvariablenames": false, "expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "prompt": "$\\int_0^1\\cos(\\frac{\\pi t}{2})\\mathrm{dt}$.
\nTo write $\\pi$ in your answer simply write pi.
", "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "answer": "2/pi", "showCorrectAnswer": true, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "showFeedbackIcon": true, "type": "jme", "scripts": {}, "showpreview": true}, {"checkvariablenames": false, "expectedvariablenames": [], "variableReplacements": [], "vsetrangepoints": 5, "prompt": "$\\int_0^1\\frac{e^z+1}{e^z+z}\\mathrm{dz}$.
\nExpress your answer using the natural log, ln().
\nHint: make a substition using the lower line.
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