5 questions on indefinite integration. Includes integration by parts and integration by substitution.
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "timing": {"timedwarning": {"message": "", "action": "none"}, "allowPause": true, "timeout": {"message": "", "action": "none"}}, "showQuestionGroupNames": true, "name": "Maria's copy of Tien Chern's copy of Henrik Skov's copy of mathcentre: Indefinite integration", "navigation": {"onleave": {"message": "", "action": "none"}, "browse": true, "reverse": true, "allowregen": true, "preventleave": false, "showresultspage": "oncompletion", "showfrontpage": false}, "showstudentname": true, "feedback": {"showanswerstate": true, "allowrevealanswer": true, "intro": "", "feedbackmessages": [], "advicethreshold": 0, "showtotalmark": true, "showactualmark": true}, "question_groups": [{"name": "Indefinite integration by substitution", "pickQuestions": 1, "pickingStrategy": "all-ordered", "questions": [{"name": "Indefinite integral", "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", "Steps", "calculus", "constant of integration", "indefinite integration", "integrals", "integrating fractional powers", "integrating powers", "integration", "standard integrals", "steps"], "advice": "\n \n \nUsing
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n \n \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n \n &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n \n \\end{eqnarray*}\\]
$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$
\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 ", "gaps": [{"notallowed": {"message": "Input all numbers as integers or fractions and not decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "type": "jme"}], "steps": [{"prompt": "The indefinite integral of a power $x^n$ where $n\\neq -1$ is \\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\]
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nIntegrate the following function $f(x)$.
\n
Input the constant of integration as $C$.
2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tChecked calculation. OK.
\n \t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n \t\tMessage about Show steps included. Also another message about including the constant of integration.
\n \t\tChanged checking range from 0 to 1 to 1 to 2.
\n \t\tImproved display.
\n \t\t", "description": "Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral", "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", "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*}\\]
$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\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 ", "gaps": [{"notallowed": {"message": "Input all numbers as integers or fractions and not decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "type": "jme"}], "steps": [{"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*}\\]
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\nIntegrate the following function $f(x)$.
\n
Input the constant of integration as $C$.
2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tCorrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.
\n \t\tChecked calculation. OK.
\n \t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n \t\tMessage about Show steps included. Also another message about including the constant of integration.
\n \t\tChanged checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.
\n \t\tImproved display of Steps by aligning integral signs.
\n \t\t", "description": "Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral", "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", "Steps", "calculus", "constant of integration", "indefinite integration", "integrals", "integration", "integration by substitution", "standard integrals", "steps", "substitution"], "advice": "\nLet $y = \\simplify[std]{{a}*x+{d}}$. 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$\\displaystyle \\int \\frac{1}{y^n} dx = -\\frac{1}{(n-1)y^{n-1}} + C$ is a standard integral, so we can now calculate the desired integral:
\n\\[\\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 ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 1.0, "prompt": "\n$\\displaystyle \\int \\simplify[std]{{b}/(({a}*x+{d})^{n})} dx= \\phantom{{}}$[[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 help. You will lose 1 mark by doing so.
\n ", "gaps": [{"notallowed": {"message": "Input all numbers as integers or fractions and not decimals.
", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.0001, "vsetrange": [0.0, 1.0], "vsetrangepoints": 5.0, "checkingtype": "reldiff", "answersimplification": "std", "marks": 3.0, "answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "type": "jme"}], "steps": [{"prompt": "\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]
", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n \n \nFind the following indefinite integral.
\n \n \n \nInput the constant of integration as $C$.
\n \n \n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(2..9)", "name": "a"}, "b": {"definition": "random(2..5)", "name": "b"}, "d": {"definition": "random(1..9)", "name": "d"}, "n": {"definition": "random(3..5)", "name": "n"}}, "metadata": {"notes": "\n \t\t2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tAdded decimal point to forbidden strings along with message to user re input of numbers.
\n \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\tChanged accuracy setting to relative difference of 0.00001 as we have negative powers.
\n \t\tChecked calculation. OK.
\n \t\tAdded message in prompt about including the constant of integration.
\n \t\tNoted issue with steps-answer order and the messages/marks generated.
\n \t\tChanged numerator to the range 2..5.
\n \t\tImproved display in Advice.
\n \t\t\n \t\t", "description": "
Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$
", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Indefinite integral", "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", "Steps", "calculus", "constant of integration", "indefinite integration", "integration", "integration by substitution", "steps", "substitution"], "advice": "\nLet $y = \\simplify[std]{{a}*x+{d}}$.
\nThen $\\displaystyle x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ and so we have the numerator $\\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$:
\n$\\displaystyle \\simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so
\n\\[\\displaystyle \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} = \\simplify[std]{({m}*y+{r})/(y^{n})={m}/y^{n-1}+{r}/y^{n}}\\]
\nNow,
\\[\\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 $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.
\nWe can now calculate the desired integral:
\n\\[ \\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}}\\]
$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\nYou 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$g(x)=\\;$[[0]]
\nRemember to input all numbers as integers or fractions.
\nClick on Show steps to get help if you need it. You will lose 1 mark by doing so.
\n ", "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": "\nFind the following indefinite integral.
\nInput all numbers as integers or fractions, not as decimals.
\nInput the constant of integration as $C$.
\n ", "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\t2/08/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \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\tChecked calculation. OK.
\n \t\tImproved display in Advice.
\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": []}]}]}], "type": "exam", "contributors": [{"name": "Tien Chern Chia", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1601/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": []}