// Numbas version: exam_results_page_options {"questions": [], "duration": 0, "name": "Integration using standard integrals", "showQuestionGroupNames": false, "allQuestions": true, "percentPass": 0, "feedback": {"showanswerstate": true, "advicethreshold": 0, "showactualmark": true, "allowrevealanswer": true, "showtotalmark": true}, "shuffleQuestions": false, "question_groups": [{"pickingStrategy": "all-ordered", "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": "\n

1. $\\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]]

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Input your answer as a fraction.


3. $\\displaystyle \\int_0^{\\pi}\\;\\cos\\left(\\frac{x}{\\var{2*n}}\\right)\\,dx=\\;$[[2]]

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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

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Input your answer as a fraction

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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\t

20/06/2012:

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Added tags.

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Improved display of prompts.

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Incomplete loading of question into editor noted. OK if refreshed.

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Changed 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": "\n

1. 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*} \\]

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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*} \\]

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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.

\n "}, {"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": {"w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "w"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol"}, "m4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2", "description": "", "name": "m4"}, "tans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(e^(p)*(p^2-2*p+2)-2)/(n4^3)", "description": "", "name": "tans4"}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1,2,3)", "description": "", "name": "b4"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "d1"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "", "name": "a1"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "m3"}, "tol1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.0001", "description": "", "name": "tol1"}, "tans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(w=0,((-1)^(m3)-1)/m3^2,-pi*(-1)^(m3)/m3)", "description": "", "name": "tans3"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "t"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "n4*b4", "description": "", "name": "p"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ln(1+b2/m2),2)", "description": "", "name": "ans2"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans3,2)", "description": "", "name": "ans3"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "b1"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans1,2)", "description": "", "name": "ans1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "name": "b2"}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tans4,4)", "description": "", "name": "ans4"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*random(1..9)", "description": "", "name": "c1"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "m2"}, "tans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1^2*(b1^5-1)/5+a1*c1*(b1^4-1)/2+(2*a1*d1+c1^2)*(b1^3-1)/3+c1*d1*(b1^2-1)+d1^2*(b1-1)", "description": "", "name": "tans1"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-1", "description": "", "name": "s6"}, "n4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1,2,3)", "description": "", "name": "n4"}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "s7"}}, "ungrouped_variables": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "d1", "s2", "s7", "s6", "m4", "m3", "m2", "tol", "a1", "tans4", "c1", "tans1", "tans3", "tol1", "p", "t", "w", "n4"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "

\\[I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\\]

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

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Input your answer to 2 decimal places.

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

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

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Input 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\\]

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

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Input your answer to 2 decimal places.

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\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]

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

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Input your answer to 4 decimal places.

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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:

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Added tags

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3/07/1012:

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Added tags.

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Checked calculations.

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

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Improved display in Advice.

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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.

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20/07/2012:

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

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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.

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Added description.

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25/07/2012:

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Added tags.

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A lot of work in this question - Perhaps it would be more managable broken down into two separate questions?

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Question appears to be working correctly.

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", "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:

\n

\\[\\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*} \\]

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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*} \\]

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c)
\\[I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\\]
We use integration by parts.

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Recall that:
\\[\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\\]
Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$

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Hence \\[v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\\]

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So \\[\\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)

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\\[I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\\]

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Use 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*} \\]

"}, {"name": "Indefinite integral of 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": {"n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sn*random(1..9)", "description": "", "name": "n"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sm*random(1..9)", "description": "", "name": "m"}, "sm": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "sm"}, "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"}}, "ungrouped_variables": ["sp", "m", "n", "p", "sn", "sm"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{n}/2*x^2 + {m} * x + {p}/2 * ln(1+x^2)+C", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

$\\displaystyle \\int f(x)\\;dx=\\;\\;$[[0]]

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Input the arbitrary constant of integration as $C$.

", "showCorrectAnswer": true, "marks": 0}], "statement": "

Integrate the following function $f(x)$

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\\[f(x)=\\simplify[std]{({n}x^3+{m}x^2+{n+p}x +{m})/(1+x^2)}\\]

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Note 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.

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Instead 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:

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Added tags.

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Checked calculation.

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19/07/2012:

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23/07/2012:

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Added tags.

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Question appears to be working correctly.

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", "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 \n

Since the degree of the numerator of $f(x)$ is greater than the denominator, $f(x)$ is improper.

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First, perform a long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{m}+({p}x)/(1+x^2)}}$.

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Each 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\t

You 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\t

Remember to input all numbers as integers or fractions.

\n\t\t\t

Click 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\t

Find the following indefinite integral.

\n\t

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

\n\t

Input 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\t

2/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\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

Checked calculation. OK.

\n\t\t

Improved 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\t

Let $y = \\simplify[std]{{a}*x+{d}}$.

\n\t

Then $\\displaystyle x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ 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\t

Now,
\\[\\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 \\]

\n\t

Since $\\displaystyle x = \\simplify[std]{(y-{d})/{a}}$ then $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.

\n\t

We 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}}\\]

\n\t"}, {"name": "Indefinite 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": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(2..8)", "description": "", "name": "c3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(3..9)", "description": "", "name": "a2"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..9)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a", "b", "s3", "s2", "s1", "s5", "s4", "a1", "a2", "b1", "c3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "gaps": [{"answer": "({b}/{a}) * e ^({a}*x) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"message": "

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]]

\n

Enter 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": "\n

Integrate the following function $f(x)$.

\n


You must input the constant of integration as $C$.

\n ", "tags": ["Calculus", "calculus", "checked2015", "exponential function", "functions", "indefinite integral", "indefinite integration", "integration", "integration of an exponential", "integration of an integer power", "integration of trigonometric functions", "mas1601", "MAS1601", "Steps", "steps", "trigonometric function"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

 20/06/2012:

\n \t\t

Added tags.

\n \t\t

Tidied up display of prompt using \\displaystyle.

\n \t\t

Problems with display of $e^{ax}$ for $a \\lt 0$. Had brackets around the $a$. (Corrected as an issue 29/06/2012).

\n \t\t

Mistake in Show steps, corrected.

\n \t\t

Added requirement to enter numbers as fractions or integers.

\n \t\t

 

\n \t\t

3/07/2012:

\n \t\t

Added tags.

\n \t\t

 

\n \t\t

9/07/2012:

\n \t\t

Extended 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": "\n

Splitting 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]]

\n

You must input the arbitrary constant of integration as $C$.

\n

Also input all numbers as fractions or integers and not as decimals.

\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Integrate the following function $f(x)$

\n

\\[f(x)=\\simplify[std]{({n}x^3+{n}x+{p})/(1+x^2)}\\]

\n

Note 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.

\n \t\t

Improved display of question prompt.

\n \t\t

Changed instructions for inputting integration constant

\n \t\t

Added decimal point . as forbidden string to stop decimal input (is this necessary?)

\n \t\t

18/07/2012:

\n \t\t

Added description.

\n \t\t

23/07/2012:

\n \t\t

Added tags.

\n \t\t

Solution always requires arctan(x) and not arcsin(x) or arccos(x). Is this on purpose?

\n \t\t

Question 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.

\n

First, perform a polynomial long division, so that $f(x)$ can be rewritten in the form $\\displaystyle{f(x)=\\simplify[std]{{n}x+{p}/(1+x^2)}}$.

\n

Each 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

$I=\\displaystyle \\int \\simplify[std]{({b}*x+{c})/(({a}*x+{d})^{n})} dx$

\n

You are given that \\[I=\\simplify[std]{g(x)*({a}x+{d})^{1-n}}+C\\] for a polynomial $g(x)$.

\n

You have to find $g(x)$.

\n

$g(x)=\\;$[[0]]

\n

Remember to input all numbers as integers or fractions.

\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n

Find the following indefinite integral.

\n

Input 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:

\n \t\t

Added tags

\n \t\t

20/06/2012:

\n \t\t

Added tags.

\n \t\t

Improved spacing and display.

\n \t\t

Got rid of instruction about including constant of integration as not needed.

\n \t\t

Checked 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": "\n

Let $y = \\simplify[std]{{a}*x+{d}}$.

\n

Then $x=\\frac{1}{\\var{a}}\\simplify[std]{(y-{d})}$ 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}}\\]

\n

Now,
\\[\\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 \\]

\n

Since $\\displaystyle x = \\simplify[std]{(y-{d})/{a}}$ then $\\displaystyle \\frac{dx}{dy} = \\frac{1}{\\var{a}}$.

\n

We 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}}\\]

\n "}, {"name": "Integration: Definite Integration", "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/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "

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

\n

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

\n

Input 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]]

\n

Input 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]]

\n

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

\n

No solutions given in Advice to parts a and c.

\n

Tolerance 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*} \\]

\n

 

"}, {"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\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 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\t

Integrate the following function $f(x)$.

\n\t


Input the constant of integration as $C$.

\n\t", "tags": ["Calculus", "calculus", "checked2015", "constant of integration", "indefinite integration", "integrating fractional powers", "integrating powers", "integration", "mas1601", "MAS1601", "standard integrals", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

2/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\t\t

Checked calculation. OK.

\n\t\t

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

\n\t\t

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

\n\t\t

Changed checking range from 0 to 1 to 1 to 2.

\n\t\t

Improved 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\t

Using
\\[\\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*}\\]

\n\t \n\t \n\t"}, {"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"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(2..8)", "description": "", "name": "c3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(3..9)", "description": "", "name": "a2"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s5"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(2..9)", "description": "", "name": "b"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s3"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s3*random(2..9)", "description": "", "name": "b1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(2..5)", "description": "", "name": "a"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s2"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s4"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a1"}}, "ungrouped_variables": ["a", "b", "s3", "s2", "s1", "s5", "s4", "a1", "a2", "b1", "c3"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0, "scripts": {}, "gaps": [{"answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+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

$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$

\n\t\t\t

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[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 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\t

Integrate the following function $f(x)$.

\n\t

 
Input the constant of integration as $C$.

\n\t", "tags": ["calculus", "Calculus", "checked2015", "constant of integration", "exponential function", "indefinite integration", "integrating powers", "integration", "integration of exponential function", "integration of powers", "integration of trigonometric functions", "mas1601", "MAS1601", "standard integrals", "Steps", "steps", "trigonometric functions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t

2/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\t\t

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

\n\t\t

Checked calculation. OK.

\n\t\t

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

\n\t\t

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

\n\t\t

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

\n\t\t

Improved 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\t

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\t

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\t"}, {"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": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "b"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "n"}}, "ungrouped_variables": ["a", "b", "d", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "(-{b})/({a*(n-1)}*({a}*x+{d})^{n-1}) + C", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

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\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", "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\t

Find the following indefinite integral.

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

Input 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\t

2/08/2012:

\n\t\t

Added tags.

\n\t\t

Added description.

\n\t\t

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

\n\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

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

\n\t\t

Checked calculation. OK.

\n\t\t

Added message in prompt  about including the constant of integration.

\n\t\t

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

\n\t\t

Changed numerator to the range 2..5.

\n\t\t

Improved 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\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"}], "name": "", "pickQuestions": 0}], "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

Questions which rely on knowledge of standard integrals.

"}, "type": "exam", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "browse": true, "showresultspage": "oncompletion", "preventleave": true, "allowregen": true, "showfrontpage": true}, "timing": {"timedwarning": {"action": "none", "message": ""}, "timeout": {"action": "none", "message": ""}, "allowPause": true}, "pickQuestions": 0, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "extensions": [], "custom_part_types": [], "resources": []}