Questions which rely on knowledge of standard integrals.
", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Definite integration using standard integrals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a1-1)*2^(a1-1)", "description": "", "name": "q"}, "val": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2*n*sin(pi/(2*n)),3)", "description": "", "name": "val"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "n"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2^(a1-1)-1", "description": "", "name": "p"}, "tolerance": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tolerance"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "9", "description": "", "name": "a1"}}, "ungrouped_variables": ["a", "a1", "val", "n", "q", "p", "tolerance"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "\n1. $\\displaystyle \\int_0^\\infty\\;e^{-\\var{a}x}\\,dx=\\; $[[0]]
Input your answer as a fraction.
2. $\\displaystyle \\int_1^2\\;\\frac{1}{x^\\var{a1}}\\,dx=\\;$[[1]]
Input your answer as a fraction.
3. $\\displaystyle \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{\\var{2*n}}\\right)\\,dx=\\;$[[2]]
Input your answer to 3 decimal places.
\n ", "scripts": {}, "gaps": [{"answer": "1/{a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "Input your answer as a fraction
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{p}/{q}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "Input your answer as a fraction
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "reldiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "val+tolerance", "minValue": "val-tolerance", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Evaluate the following definite integrals.
", "tags": ["calculus", "Calculus", "checked2015", "definite integral", "definite integration", "exponential function", "integration", "integration of a negative power", "integration of a negative power ", "integration of an exponential", "integration of trigonometric functions", "mas1601", "MAS1601", "trigonometric functions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved display of prompts.
\n \t\tIncomplete loading of question into editor noted. OK if refreshed.
\n \t\tChanged accuracy for second question to relative difference of 0.0001 to ensure it marked correctly for extreme values.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Calculate definite integrals: $\\int_0^\\infty\\;e^{-ax}\\,dx$, $\\int_1^2\\;\\frac{1}{x^{b}}\\,dx$, $\\; \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{2n}\\right)\\,dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n1. We have
\\[\\int\\;e^{-\\var{a}x}\\,dx=-\\frac{1}{\\var{a}}e^{-\\var{a}x} +C\\]
Also we know that $\\lim_{x \\to \\infty}\\;e^{-\\var{a}x}=0$.
Hence:
\\[\\begin{eqnarray*} \\int_0^\\infty\\;e^{-\\var{a}x}\\,dx&=&-\\frac{1}{\\var{a}}\\left[e^{-\\var{a}x}\\right]_0^\\infty\\\\ &=&-\\frac{1}{\\var{a}}\\left(\\left(\\lim_{x \\to \\infty}e^{-\\var{a}x}\\right)-1\\right)\\\\ &=&-\\frac{1}{\\var{a}}(0-1)\\\\ &=&\\frac{1}{\\var{a}} \\end{eqnarray*} \\]
2. \\[\\int\\;\\frac{1}{x^\\var{a1}}\\,dx= \\simplify[std]{x^{-a1+1}/{-a1+1}}+C\\]
Hence:
\\[ \\begin{eqnarray*}\\int_1^2\\;\\frac{1}{x^\\var{a1}}\\,dx&=&\\left[\\simplify[std]{x^{-a1+1}/{-a1+1}}\\right]_1^2\\\\&=& -\\simplify[std]{{1}/{a1-1}}\\left(2^{\\var{-a1+1}}-1\\right)\\\\&=&\\simplify[std]{{p}/{q}} \\end{eqnarray*} \\]
3. \\[ \\begin{eqnarray*} \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{\\var{2*n}}\\right)\\,dx&=&\\var{2*n}\\left[\\sin\\left(\\frac{x}{\\var{2*n}}\\right)\\right]_0^{\\pi}\\\\ &=&\\var{2*n}\\left(\\sin\\left(\\frac{\\pi}{\\var{2*n}}\\right)-0\\right)\\\\ &=&\\var{val} \\end{eqnarray*} \\]
to 3 decimal places.
\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 2 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 2 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 2 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans3+tol", "minValue": "ans3-tol", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 4 decimal places.
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "ans4+tol1", "minValue": "ans4-tol1", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Evaluate the following definite integrals.
", "tags": ["Calculus", "calculus", "checked2015", "definite integration", "integration", "integration by parts", "integration by parts twice", "MAS1601", "mas1601"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "15/07/2015:
Added tags
\n3/07/1012:
\nAdded tags.
\nChecked calculations.
\nLeft tolerances in, as easy to make minor errors in calculations.
\nImproved display in Advice.
\nSome 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.
\n20/07/2012:
\nSet new tolerace variables, tol=0.01, tol1=0.0001.
\nCan 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.
\nAdded description.
\n\n25/07/2012:
\n\n
Added tags.
\nA lot of work in this question - Perhaps it would be more managable broken down into two separate questions?
\n\n
Question appears to be working correctly.
\n\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "
Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
\\[I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]
Expand the parentheses to obtain:
\\[\\begin{eqnarray*}I &=& \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &=&\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &=&\\var{tans1}\\\\ \\\\&=&\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]
\nb)
\\[\\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*} \\]
c)
\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
We use integration by parts.
Recall that:
\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]
Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$
Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]
\nSo \\[\\begin{eqnarray*} I&=&\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &=&\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &=& \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \\]
d)
\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]
\nUse integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$
\\[\\begin{eqnarray*} I&=&\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &=&\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &=&\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \\]
$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]
\nInput the arbitrary constant of integration as $C$.
", "showCorrectAnswer": true, "marks": 0}], "statement": "Integrate the following function $f(x)$
\n\\[f(x)=\\simplify[std]{({n}x^3+{m}x^2+{n+p}x +{m})/(1+x^2)}\\]
\nNote that if you need to enter the absolute value in your answer, e.g. $|x|$, then you should not use the vertical bar on the keyboard.
\nInstead you must use the abs() function, i.e. abs(x).
", "tags": ["Calculus", "checked2015", "degree of polynomial", "indefinite integration", "integration", "logarithm", "logs", "long division of polynomials", "MAS1601", "polynomial division", "polynomials"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "2/07/2012:
\nAdded tags.
\nChecked calculation.
\n19/07/2012:
\nAdded description.
\n23/07/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle \\int \\frac{nx^3+mx^2+px +m}{x^2+1} \\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n \nSince the degree of the numerator of $f(x)$ is greater than the denominator, $f(x)$ is improper.
\n \n \n \nFirst, perform a long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{m}+({p}x)/(1+x^2)}}$.
\n \n \n \nEach term of this expression can then be integrated using standard functions (to within the arbitrary constant) to give:
\n \n \n \n$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{{n}x^2/2+{m}x+{p}/2*ln(1+x^2)} +C}$
\n \n "}, {"name": "Indefinite integral: polynomial fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m*a", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m*d+r", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "d"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({-m}/{n-2})*x-{m*d*(n-1)+r*(n-2)}/{(n-2)*(n-1)*a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\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", "steps": [{"type": "information", "prompt": "One way to do this is by substitution, for example $y = \\simplify[std]{{a}*x+{d}}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "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", "tags": ["Calculus", "MAS1601", "Steps", "checked2015", "constant of integration", "indefinite integration", "integration", "integration by substitution", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the polynomial $g(x)$ such that $\\displaystyle \\int \\frac{ax+b}{(cx+d)^{n}} dx=\\frac{g(x)}{(cx+d)^{n-1}}+C$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\tLet $y = \\simplify[std]{{a}*x+{d}}$.
\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}}\\]
Enter all numbers as integers or fractions and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]
\nEnter all numbers as integers or fractions and not as decimals.
\n ", "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*}\\]
", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 0}], "statement": "\nIntegrate the following function $f(x)$.
\n
You must input the constant of integration as $C$.
20/06/2012:
\n \t\tAdded tags.
\n \t\tTidied up display of prompt using \\displaystyle.
\n \t\tProblems with display of $e^{ax}$ for $a \\lt 0$. Had brackets around the $a$. (Corrected as an issue 29/06/2012).
\n \t\tMistake in Show steps, corrected.
\n \t\tAdded requirement to enter numbers as fractions or integers.
\n \t\t\n \t\t
3/07/2012:
Added tags.
\n \t\t\n \t\t
9/07/2012:
\n \t\tExtended ruleset std to include !noLeadingMinus so that answer is displayed in the right order.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Integrate $f(x) = ae ^ {bx} + c\\sin(dx) + px^q$. Must input $C$ as the constant of integration.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nSplitting the integral into three parts and using the information in Steps we have:
\n\\[\\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 "}, {"name": "Integrate algebraic fraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"sp": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sp"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sp*random(1..9)", "description": "", "name": "p"}, "sn": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sn"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sn*random(1..9)", "description": "", "name": "n"}}, "ungrouped_variables": ["p", "sp", "sn", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "((({n} * (x ^ 2)) / 2) + ({p} * Arctan(x))+C)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as fractions or integers and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]
\nYou must input the arbitrary constant of integration as $C$.
\nAlso input all numbers as fractions or integers and not as decimals.
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nIntegrate the following function $f(x)$
\n\\[f(x)=\\simplify[std]{({n}x^3+{n}x+{p})/(1+x^2)}\\]
\nNote that you can only enter inverse trigonometric functions as $\\arcsin(x),\\;\\;\\arccos(x),\\;\\;\\arctan(x)$.
\n\n ", "tags": ["Calculus", "MAS1601", "arctan", "checked2015", "degree of a polynomial", "improper rational polynomials", "indefinite integration", "integration", "integration of standard functions", "integration using trigonometric identities", "inverse trigonometric functions", "polynomial division", "trigonometric identities"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t
28/06/2012:
\n \t\t
Added tags.
Improved display of question prompt.
\n \t\tChanged instructions for inputting integration constant
\n \t\tAdded decimal point . as forbidden string to stop decimal input (is this necessary?)
\n \t\t18/07/2012:
\n \t\tAdded description.
\n \t\t23/07/2012:
\n \t\tAdded tags.
\n \t\tSolution always requires arctan(x) and not arcsin(x) or arccos(x). Is this on purpose?
\n \t\tQuestion appears to be working correctly.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int\\frac{ax^3+ax+b}{1+x^2}\\;dx$. Enter the constant of integration as $C$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Since the degree of the numerator of $f(x)$ is greater than the denominator, $f(x)$ is improper.
\nFirst, perform a polynomial long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{p}/(1+x^2)}}$.
\nEach term of this expression can then be integrated using standard functions (to within the arbitrary constant) to give:
\n$\\displaystyle{\\int f(x)\\;dx=\\simplify[std]{{n}x^2/2+{p}arctan(x)} +C}$
"}, {"name": "Integration of fraction with power in denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "3", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m*a", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "m*d+r", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "d"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..4)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "({-m}/{n-2})*x-{m*d*(n-1)+r*(n-2)}/{(n-2)*(n-1)*a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$
\nYou are given that \\[I=\\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C\\] for a polynomial $g(x)$.
\nYou have to find $g(x)$.
\n$g(x)=\\;$[[0]]
\nRemember to input all numbers as integers or fractions.
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nFind the following indefinite integral.
\nInput all numbers as integers or fractions, not as decimals.
\n\n ", "tags": ["Calculus", "calculus", "checked2015", "indefinite integral", "indefinite integration", "integration", "integration by substitution", "integration of a rational polynomial", "MAS1601", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t
3/7/2012:
Added tags
\n \t\t20/06/2012:
\n \t\tAdded tags.
\n \t\tImproved spacing and display.
\n \t\tGot rid of instruction about including constant of integration as not needed.
\n \t\tChecked calculation.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "$\\displaystyle \\int \\frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1-n}+C$ for a polynomial $g(x)$. Find $g(x)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\nLet $y = \\simplify[std]{{a}*x+{d}}$.
\nThen $x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ and so we have the numerator $\\simplify[std]{{b}*x+{c}}$ becomes in terms of $y$:
\n$\\simplify[std]{{b}*x+{c} = {b}*1/{a}*(y-{d})+{c}= {m}y+{r}}$ and so
\n\\[\\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=\\int_0^{\\var{b1}}\\simplify[std]{e^({a}x)}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 3 decimal places.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans1-tol", "maxValue": "ans1+tol", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\\[I=\\int_0^{\\var{b2}}\\simplify[std]{1/({b}x+{m2})}\\;dx\\]
\n$I=\\;\\;$[[0]]
\nInput your answer to 3 decimal places.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans2-tol", "maxValue": "ans2+tol", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "\\[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.
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "minValue": "ans3-tol", "maxValue": "ans3+tol", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"w": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(0,1)", "name": "w", "description": ""}, "tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "name": "tol", "description": ""}, "ans2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(1/b*(ln(1+b*b2/m2)),3)", "name": "ans2", "description": ""}, "ans3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(tans3,3)", "name": "ans3", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..2#0.5 except 0)", "name": "a", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1..2#0.5 except 0)", "name": "b1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "name": "d1", "description": ""}, "s2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "s2", "description": ""}, "ans1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(tans1,3)", "name": "ans1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..20)", "name": "b2", "description": ""}, "m3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "m3", "description": ""}, "tol1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.0001", "name": "tol1", "description": ""}, "tans3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "1/m3*((1-w)*sin(m3*pi/2)-w*(cos(m3*pi/2)-1))", "name": "tans3", "description": ""}, "m2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "m2", "description": ""}, "tans1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(1/a)*(e^(a*b1)-1)", "name": "tans1", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "name": "b", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "t*random(1..9)", "name": "c1", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "name": "t", "description": ""}}, "ungrouped_variables": ["a", "b", "m3", "s2", "ans2", "ans3", "b2", "ans1", "tol", "b1", "w", "m2", "c1", "d1", "tans1", "tol1", "tans3", "t"], "functions": {}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Evaluate the following definite integrals.
", "tags": ["Calculus", "calculus", "checked2015", "definite integration", "integration"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Evaluate $\\int_0^{\\,m}e^{ax}\\;dx$, $\\int_0^{p}\\frac{1}{bx+d}\\;dx,\\;\\int_0^{\\pi/2} \\sin(qx) \\;dx$.
\nNo solutions given in Advice to parts a and c.
\nTolerance of 0.001 in answers to parts a and b. Perhaps should indicate to the student that a tolerance is set. The feedback on submitting an incorrect answer within the tolerance says that the answer is correct - perhaps there should be a different feedback in this case if possible for all such questions with tolerances.
"}, "advice": "\n
b)
\\[\\begin{eqnarray*}I&=&\\int_0^{\\var{b2}}\\simplify[std]{1/({b}*x+{m2})}\\;dx\\\\ &=&\\frac{1}{\\var{b}}\\left[\\ln(\\var{b}x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &=&\\frac{1}{\\var{b}}\\left\\{ \\ln(\\var{b2*b+m2})-\\ln(\\var{m2})\\right\\}\\\\ &=&\\frac{1}{\\var{b}}\\ln\\left(\\frac{\\var{b2*b+m2}}{\\var{m2}}\\right)\\\\ &=&\\var{ans2}\\mbox{ to 3 decimal places} \\end{eqnarray*} \\]
"}, {"name": "Integration: Indefinite integral", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "r": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..b-1)", "description": "", "name": "r"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*b+r", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "name": "c"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "s1", "m", "n", "r"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({c}/{m+1})x ^ {m+1} + ({d*n}/{b+n})*x^({n+b}/{n})+C", "vsetrange": [1, 2], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "
Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$
\n\t\t\t$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[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 more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"type": "information", "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\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
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", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\t \n\t \n\tUsing
\\[\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\\] for any number $n \\neq -1$ we have
\\[\\begin{eqnarray*}\n\t \n\t \\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\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\\]
Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$
\n\t\t\t$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[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 more information. You will not lose any marks by doing so.
\n\t\t\t", "steps": [{"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*}\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tIntegrate the following function $f(x)$.
\n\t
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", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\tNote 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\tSplitting 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*}\\]
Input all numbers as integers or fractions and not decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "reldiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "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", "steps": [{"type": "information", "prompt": "\\[\\int (ax+b)^n \\;dx = \\frac{1}{a(n+1)}(ax+b)^{n+1}+C\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n\t \n\t \n\tFind the following indefinite integral.
\n\t \n\t \n\t \n\tInput the constant of integration as $C$.
\n\t \n\t \n\t", "tags": ["calculus", "Calculus", "checked2015", "constant of integration", "indefinite integration", "integration", "integration by substitution", "MAS1601", "mas1601", "standard integrals", "Steps", "steps", "substitution"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "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", "licence": "Creative Commons Attribution 4.0 International", "description": "
Find $\\displaystyle \\int \\frac{a}{(bx+c)^n}\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"}]}], "contributors": [{"name": "Henrik Skov Midtiby", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/829/"}], "extensions": [], "custom_part_types": [], "resources": []}