// Numbas version: exam_results_page_options {"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Integration by Substitution

\n

rebelmaths

\n

rebel

"}, "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

\$I=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\$

\n

$I=\\;\\;$[[0]]

\n

\n ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans1-tol", "marks": 1.0, "maxvalue": "ans1+tol"}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n

\$I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\$

\n

$I=\\;\\;$[[0]]

\n

\n ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans2-tol", "marks": 1.0, "maxvalue": "ans2+tol"}], "marks": 0.0, "type": "gapfill"}, {"prompt": "\n

\$I=\\int_0^{\\pi/2}\\simplify[std]{({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\$

\n

$I=\\;\\;$[[0]]

\n

\n ", "gaps": [{"showPrecisionHint": false, "type": "numberentry", "minvalue": "ans3-tol", "marks": 1.0, "maxvalue": "ans3+tol"}], "marks": 0.0, "type": "gapfill"}], "statement": "

Evaluate the following definite integrals.

", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n \t\t \t\t

3/07/1012:

\n \t\t \t\t

\n \t\t \t\t

Checked calculations.

\n \t\t \t\t

Left tolerances in, as easy to make minor errors in calculations.

\n \t\t \t\t

\n \t\t \t\t

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.

\n \t\t \t\t

20/07/2012:

\n \t\t \t\t

Set new tolerace variables, tol=0.01, tol1=0.0001.

\n \t\t \t\t

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.

\n \t\t \t\t

\n \t\t \t\t

\n \t\t \t\t

25/07/2012:

\n \t\t \t\t

\n \t\t \t\t

\n \t\t \t\t

A 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 $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.

\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

\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": "

Integration by substitution. Hint given on susbtitution

\n

rebelmaths

"}, "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": "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

$\\displaystyle f(x)=\\simplify[std]{{a}*x^({b}/{c})}$

\n

Input $\\displaystyle \\int f(x)\\;dx$ here.

\n

\n ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "

Input all numbers as integers or fractions and not as decimals.

"}}, {"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

$f(x)=\\simplify[std]{{t[0]}*sin({b}x+{c})+{t[1]}*cos({b}x+{c})+{t[2]}*exp({b}x+{c})}$

\n

Input $\\displaystyle \\int f(x)\\;dx$ here.

\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

$f(x)=\\simplify[std]{{a}exp({b}/{c}*x)}$

\n

Input $\\displaystyle \\int f(x)\\;dx$ here.

\n ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "

Input all numbers as integers or fractions and not as decimals.

"}}, {"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

$\\displaystyle f(x)=\\simplify[std]{{a1}/({b1}x+{c1})}$

\n

Input $\\displaystyle \\int f(x)\\;dx$ here.

\n

\n ", "notallowed": {"strings": ["."], "partialcredit": 0.0, "showstrings": false, "message": "

Input all numbers as integers or fractions and not as decimals.

"}}], "type": "question", "question_groups": [{"questions": [], "pickingStrategy": "all-ordered", "pickQuestions": 0, "name": ""}], "progress": "ready", "tags": [], "metadata": {"notes": "", "description": "

Integrating simple functions.

", "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

Integrate the following functions $f(x)$.

\n

Input all numbers as integers or fractions and not as decimals.

\n

In all examples do not forget to include the constant of integration $C$.

\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": "

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": "

$\\int\\frac{\\mathrm{dx}}{\\var{a}x+\\var{b}}$.

\n

Answer 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": "

$\\int \\frac{x \\mathrm{dx}}{\\var{c}+x^2}$.

\n

Answer 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\t

This 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 $\\frac{1}{\\var{2*a}}du$.

\n\t

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\t

A 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 $x$.

\n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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\t

Input 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\t

Find the following integral.

\n\t

Input 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\t

2/08/2012:

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

Checked calculation. OK.

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

Added decimal point as forbidden string and included message in prompt about not entering decimals.

\n\t\t \t\t \t\t

Got 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\t

2/08/2012:

\n\t\t \t\t

\n\t\t \t\t

\n\t\t \t\t

Corrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.

\n\t\t \t\t

Checked calculation. OK.

\n\t\t \t\t

Added decimal point to forbidden strings along with message to user re input of numbers.

\n\t\t \t\t

Message about Show steps included. Also another message about including the constant of integration.

\n\t\t \t\t

Changed checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.

\n\t\t \t\t

Improved 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]]

\n

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

\n

Click 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": "\n

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*}\$

\n

Splitting 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*}\$

\n \n", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "variable_groups": [], "statement": "\n

Integrate the following function $f(x)$.

\n

Input the constant of integration as $C$.

\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

Find the following indefinite integral.

\n\t \n\t \n\t \n\t

Input the constant of integration as $C$.

Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$

", "notes": "\n\t\t \t\t

2/08/2012:

\n\t\t \t\t

\n\t\t \t\t

\n\t\t \t\t

Added decimal point to forbidden strings along with message to user re input of numbers.

\n\t\t \t\t

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.

\n\t\t \t\t

Changed accuracy setting to relative difference of 0.00001 as we have negative powers.

\n\t\t \t\t

Checked calculation. OK.

\n\t\t \t\t

\n\t\t \t\t

Noted issue with steps-answer order and the messages/marks generated.

\n\t\t \t\t

Changed numerator to the range 2..5.

\n\t\t \t\t

\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\t

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

\n\t\t\t

Click 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\t

Let $y = \\simplify[std]{{a}*x+{d}}$. Then,
\$\\simplify[std]{{b}/(({a}*x+{d})^{n})} = \\simplify[std]{{b}/(y^{n})}.\$

\n\t

Now,
\$\\int \\simplify[std]{{b}/({a}*x+{d})^{n}} dx = \\int \\simplify[std]{{b}/(y^{n})} \\frac{dx}{dy} dy.\$

\n\t

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\t

2/08/2012:

\n\t\t \t\t

\n\t\t \t\t

\n\t\t \t\t

Checked calculation. OK.

\n\t\t \t\t

\n\t\t \t\t

\n\t\t \n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Find $\\displaystyle \\int \\sin(x)(a+ b\\cos(x))^{m}\\;dx$

"}, "showQuestionGroupNames": false, "advice": "\n\t \n\t \n\t

This 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 $\\frac{1}{\\var{-b}}\\;du$.

\n\t \n\t \n\t \n\t

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\t

Input all numbers as integers or fractions.

\n\t\t\t

$I=\\;$[[0]]

\n\t\t\t

Input the constant of integration as $C$.

\n\t\t\t

Click 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\t

Find the following integral.

\n\t

Input the constant of integration as $C$.

\n\t

Input 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\t

2/08/2012:

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

Checked calculation. OK.

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

Added decimal point as forbidden string and included message in prompt about not entering decimals.

\n\t\t \t\t \t\t

\n\t\t \t\t \t\t

Note 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 $\\displaystyle \\int \\frac{2ax + b}{ax ^ 2 + bx + c}\\;dx$

This exercise is best solved by using substitution.

\n\t

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 $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$

\n\t

So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$

\n\t

Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$

\n\t

Hence 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*}\$

\n\t

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\$

\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

\$I=\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\$

\n\t\t\t

$I=\\;$[[0]]

\n\t\t\t

Input the constant of integration as $C$.

\n\t\t\t

Input all numbers as integers or fractions not as decimals.

\n\t\t\t

Click 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\t

Find the following integral.

\n\t

Input the constant of integration as $C$.

\n\t

Input all numbers as integers or fractions.

\n\t

\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

Find the following integral.

\n\t

Input the constant of integration as $C$.

\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

\$I=\\simplify[std]{Int( x*({a} * x ^ 2 + {b})^{m},x)}\$

\n\t\t\t

$I=\\;$[[0]]

\n\t\t\t

Input numbers in your answer as integers or fractions and not as decimals.

\n\t\t\t

Click on Show steps to get further help. You will lose 1 mark if you do so.

\n\t\t\t \n\t\t\t", "type": "gapfill", "steps": [{"type": "information", "marks": 0.0, "prompt": "

Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + {b}}$

"}], "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": "

Input all numbers as integers or fractions and not as decimals.

", "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

This 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 $\\frac{1}{\\var{2*a}}du$.

\n\t \n\t \n\t \n\t

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\t

A 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 $x$.

\n\t \n\t \n\t \n\t", "showQuestionGroupNames": false, "metadata": {"description": "

Find $\\displaystyle \\int x(a x ^ 2 + b)^{m}\\;dx$

", "notes": "\n\t\t \t\t

2/08/2012:

\n\t\t \t\t

\n\t\t \t\t

\n\t\t \t\t

Checked calculation. OK.

\n\t\t \t\t

\n\t\t \t\t

Added decimal point as forbidden string and included message in prompt about not entering decimals.

\n\t\t \t\t

Got 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"}, "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": "

$\\int_0^1\\cos(\\frac{\\pi t}{2})\\mathrm{dt}$.

\n

To write $\\pi$ in your answer simply write pi.

", "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": "

$\\int_0^1\\frac{e^z+1}{e^z+z}\\mathrm{dz}$.

\n

\n

Hint: make a substition using the lower line.

", "checkingtype": "absdiff"}], "tags": [], "metadata": {"description": "

Definite Integrals

", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "ungrouped_variables": [], "rulesets": {}, "variables": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "advice": "

Definite Integrals

", "functions": {}, "statement": "

Find the following definite integrals

", "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}, "showQuestionGroupNames": false, "type": "exam", "contributors": [{"name": "Tien Chern Chia", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1601/"}], "extensions": [], "custom_part_types": [], "resources": []}