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$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*cos({c}x+{d})} dx $
\n$I=\\;$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\nYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n ", "steps": [{"type": "information", "prompt": "\n \n \nThe 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": 4, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\nUse the result from the first part to find:
\n$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*sin({c}*x+{d})} dx $
\n$I=\\;$[[0]]
\nInput all numbers as fractions or integers and not decimals.
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\nFind the following indefinite integrals.
\nInput all numbers as fractions or integers and not decimals.
\nInput the constant of integration as $C$.
\n ", "tags": ["Calculus", "MAS1601", "Steps", "checked2015", "constant of integration", "indefinite integration", "integrating", "integrating trigonometric functions", "integration", "integration by parts", "twice"], "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\tGot rid of redundant ruleset, added !noLeadingMinus to std ruleset as we need to keep the standard order for integrating by parts.
\n \t\tChecked calculation. OK.
\n \t\tPenalised use of steps in first part, 1 mark. Added message to that effect in first part.
\n \t\tAdded message about not inputting decimals in appropriate places.
\n \t\tChanged marks reflecting the use of steps and degree of difficulty in second part.
\n \t\tImproved Advice display.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int (ax+b)\\cos(cx+d)\\; dx $ and hence find $\\displaystyle \\int (ax+b)^2\\sin(cx+d)\\; dx $
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\na)
\nThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{({a}x+{b})}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{cos({c}*x+{d})}$.
\nSo $\\displaystyle \\frac{du}{dx} = \\simplify[std]{{a}}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*sin({c}*x+{d})}$.
\nHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*cos({c}*x+{d})} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{(({a}*x+{b})/{c})*sin({c}*x+{d}) - ({a}/{c})*Int(sin({c}*x+{d}),x)} \\\\ &=& \\simplify[std]{(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}) + C} \\end{eqnarray} \\]
b)
\nFor this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({c}*x+{d})}$.
\nSo $\\displaystyle \\frac{du}{dx}=\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{-(1/{c})*cos({c}*x+{d})}$.
\nHence,
\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) + (1/{c})*Int({2*a}*({a}x+{b})*cos({c}*x+{d}),x)} \\\\ &=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*Int(({a}x+{b})*cos({c}*x+{d}),x)}\\dots (*) \\end{eqnarray*}\\]
But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*cos({c}*x+{d}),x)=(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d})}$
\nSo on substituting this in equation (*) we find:
\\[ \\begin{eqnarray*}I&=& \\simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*((({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}))+C}\\\\ &=& \\simplify[std]{-(({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C} \\end{eqnarray*}\\]