Integration: By Parts 5

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Ben McGovern 2 years, 7 months ago

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Ben McGovern 4 years, 9 months ago

Created this as a copy of Integration: By Parts 6.

Name	Status	Author	Last Modified
Integration: By Parts 1a	Ready to use	Ben McGovern	26/09/2022 12:22
Integration: By Parts 1b	Ready to use	Ben McGovern	26/09/2022 12:23
Integration: By Parts 4a	Ready to use	Ben McGovern	26/09/2022 12:25
Integration: By Parts 4b	Ready to use	Ben McGovern	26/09/2022 12:25
Integration: By Parts 2a	Ready to use	Ben McGovern	26/09/2022 12:23
Integration: By Parts 2b	Ready to use	Ben McGovern	26/09/2022 12:24
Integration: By Parts 3	Ready to use	Ben McGovern	26/09/2022 12:24
Integration: By Parts 6	Ready to use	Ben McGovern	26/09/2022 12:26
Integration: By Parts 5	Ready to use	Ben McGovern	26/09/2022 12:26
CD4 Integration - Parts	Ready to use	Andrew Neate	17/04/2025 09:02

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Question statement

Calculate the integral

$\simplify{int(ln({a}x) ,x)}$

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To calculate the integral

$\simplify{int(ln({a}x),x)},$ despite there only being a single function of

$x$ , we can make use of integration by parts. This is because there is no 'standard' integral for

$\simplify{ln({a}x)}$ .

However, if we think of

$\simplify{int(ln({a}x),x)} = \int 1 \times \ln(\var{a}x)\, \text{d}x ,$

we can set

$u=\simplify{ln({a}x)}, \quad \frac{dv}{dx}=1,$

so we only have to differentiate the $\simplify{ln({a}x)}$ term.

Now, calculating the $\tfrac{du}{dx}$ and $v(x)$ terms:

$\begin{split} u(x) = \simplify{ln({a}x)} \quad &\implies \frac{du}{dx} = \simplify{1/x}; \\ \frac{dv}{dx} = 1 &\implies v(x) = x. \end{split}$

Plugging these 4 terms into the integration by parts formula we can succesfully find a solution:

$\begin{split} \simplify{int(ln({a}x),x)} &\,= \simplify{x ln({a}x)} - \simplify[alwaysTimes]{int((1/x)x,x)}, \\ &\,=\simplify{x ln({a}x)-int(1,x)}, \\ &\,= \simplify{x ln({a}x) - x}+c, \\ &\,=\simplify{x (ln({a}x)-1)}+c.\end{split}$

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This question is used in the following exams:

Integration by Parts by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.
Integration by part by Jinhua Mathias in Maths 3-2020.
examenIntegralesPartes by Ricardo Mateos in Matemáticas II.

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Integration: By Parts 5

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Taxonomy: mathcentre

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Ben McGovern 2 years, 7 months ago

Ben McGovern 4 years, 1 month ago

Ben McGovern 4 years, 9 months ago

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