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": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\tYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n\t\t\t", "steps": [{"type": "information", "prompt": "\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
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": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n\t\t\t$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tInput the constant of integration as $C$.
\n\t\t\t", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a2}/{b}*x+{a1}/{b^2}", "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}, {"answer": "{-a1}/{b}*x+{a2}/{b^2}", "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\t\t\tUsing the first two parts find:
$\\displaystyle I=\\int \\simplify[std]{{a1}x*sin({b}x+{c})+{a2}x*cos({b}x+{c})} dx $
You are given that \\[I=\\simplify[std]{f(x)*sin({b}x+{c})+g(x)*cos({b}x+{c})+C}\\]
where $f(x)$ and $g(x)$ are polynomials of degree 1. You have to find $f(x)$ and $g(x)$.
$f(x)=\\;$[[0]] $\\;\\;\\;\\;\\;g(x)=\\;$[[1]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\t", "showCorrectAnswer": true, "marks": 0}], "statement": "\n\tFind the following indefinite integrals.
\n\tInput all numbers as fractions or integers and not decimals.
\n\tInput the constant of integration as $C$ where needed.
\n\t", "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "constant of integration", "integrating trigonometric functions", "integration", "integration by parts"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t3/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tCorrected error in second question answer, + changed to -. Also solution to second gap in third part. Advice changed accordingly.
\n\t\tChecked calculations after corrections. OK.
\n\t\tPenalised use of steps in first part, 1 mark. Added message to that effect.
\n\t\tChanged marks to allow for steps penalty.
\n\t\tImproved Advice display.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx $ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n\ta)
\n\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{{a}x}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \\int \\left( \\simplify[std]{(-{a}/{b})*cos({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \\end{eqnarray} \\]
b)
\n\tFor this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \\int \\left( \\simplify[std]{({a}/{b})*sin({b}x+{c})}\\right) dx \\\\ &=& \\simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \\end{eqnarray} \\]
c)
\n\tUsing the results from Parts a and b, we have \\[\\begin{eqnarray*}I &=& \\int \\simplify[std]{{a1}x*sin({b}x+{c})} dx + \\int \\simplify[std]{{a2}x*cos({b}x+{c})} dx\\\\ &=& \\simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\\\ &=&\\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\\\ &=&\\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \\end{eqnarray*}\\]
Hence
$\\displaystyle \\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$
$\\displaystyle \\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$
\n\t", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}