Jo-Ann's copy of Integration by parts

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Question 1

Find the following indefinite integral.

This question is scaffolded - i.e. it takes you through answering the question step by step.

Input all numbers as fractions or integers and not decimals.

Don't forget $C$ !

a)

$I=\displaystyle \int \simplify[std]{({a}x)*ln({c}x)} dx$

The formula for integration by parts is

$\int u\frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx.$

What is the most suitable choice for $u$ and $\frac{dv}{dx}$ ?

$u =\;$ interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\ln \left ( 4 x \right )}$

$\frac{dv}{dx} =\;$ interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{5 x}$

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b)

Hence find:

$\frac{du}{dx} =\;$ interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 1 }{ x }}$

$v =\;$ interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 5 x^{ 2 } }{ 2 }}$

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c)

Hence find:

$uv =\;$ interpreted as $\displaystyle{}$ Expected answer:interpreted as $\displaystyle{\frac{ 5 x^{ 2 } }{ 2 } \ln \left ( 4 x \right )}$

$\int v\frac{du}{dx}\mathrm{d}x = \;$ $+C$

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d)

Use the results from above to find:

$I=\displaystyle \int \simplify[std]{({a}x)*ln({c}x)} dx = uv - \int v \frac{du}{dx} dx = \;$ $+C$

Input all numbers as fractions or integers and not decimals.

Score: 0/2

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Advice

The formula for integrating by parts is

$\int u\frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx.$

We choose $u = \simplify[std]{ln{c}x}$ and $\displaystyle \frac{dv}{dx} = \simplify[std]{{a}x}$ .

So $\displaystyle \frac{du}{dx} = \simplify[std]{1/x}$ and $\displaystyle v = \simplify[std]{({a}x^2/2)}$ .

Hence,
$\begin{eqnarray} \int \simplify[std]{({a}*x)*ln({c}*x)} dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{(({a}*x^2)/2)*ln({c}*x) - Int(({a}*x/2),x)} \\ &=& \simplify[std]{(({a}x^2)/2)*ln({c}*x) -({a}x^2/4) + C} \end{eqnarray}$

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Jo-Ann's copy of Integration by parts - logarithm

Question 1

a)

b)

c)

d)

Advice

The formula for integrating by parts is