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Find the following indefinite integrals.

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$ where needed.

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Find $\\displaystyle \\int x\\sin(cx+d)\\;dx,\\;\\;\\int x\\cos(cx+d)\\;dx $ and hence $\\displaystyle \\int ax\\sin(cx+d)+bx\\cos(cx+d)\\;dx$

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a)

The 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 $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{sin({b}x+{c})}$.

So $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $v = \\simplify[std]{(-1/{b})*cos({b}x+{c})}$.

Hence,
\\[ \\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)

For this part we choose $u = \\simplify[std]{{a}x}$ and $\\frac{dv}{dx} = \\simplify[std]{cos({b}x+{c})}$.

So $\\displaystyle \\frac{du}{dx}$ = $\\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{b})*sin({b}x+{c})}$.

Hence,
\\[ \\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)

Using 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
$\\displaystyle \\simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$

$\\displaystyle \\simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$

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$\\displaystyle \\int \\simplify[std]{{a}x*sin({b}x+{c})} dx = \\phantom{{}}$[[0]]

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$.

You can get help by clicking on Show steps. You will lose 1 mark if you do.

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Do not input numbers as decimals, only as integers without the decimal point, or fractions

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The formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]

\n \n \n ", "useCustomName": false, "unitTests": [], "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "variableReplacements": [], "type": "information", "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showFeedbackIcon": true, "scripts": {}}], "marks": 0, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "customName": "", "showFeedbackIcon": true, "stepsPenalty": 1, "scripts": {}}, {"adaptiveMarkingPenalty": 0, "prompt": "\n

$\\displaystyle \\int \\simplify[std]{{a}x*cos({b}x+{c})} dx = \\phantom{{}}$[[0]]

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$.

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Do not input numbers as decimals, only as integers without the decimal point, or fractions

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Using 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]]

Input all numbers as fractions or integers and not decimals.

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Do not input numbers as decimals, only as integers without the decimal point, or fractions

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