Integration: By Parts 3

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Ben McGovern

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Ben McGovern 2 years, 7 months ago

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Ben McGovern 4 years, 1 month ago

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Ben McGovern 4 years, 9 months ago

Created this as a copy of Integration: By Parts 2b.

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

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

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Advice

If we have a function of $x$ which is the product of two functions of $x$ , to integrate such a function it is often necessary to use Integration by Parts. The formula for Integration by Parts is:

$\int u(x) \frac{dv}{dx} dx = u(x)v(x) - \int v(x) \frac{du}{dx} dx.$

Using this method can be broken down into steps:

Identify $u(x)$ and $\tfrac{dv}{dx}$ (The function you pick for each is important, in general you want $u(x)$ to become simpler when differentiating it, and you must be able to integrate $\tfrac{dv}{dx}$ to find $v(x)$ );
Calculate $\tfrac{du}{dx}$ and $v(x)$ ;
Put the functions $u(x)$ , $v(x)$ , and their derivatives into the Integration by Parts formula;
Calculate the integral $\int v(x) \tfrac{du}{dx} dx$ (This may require you to use Integration by Parts again, this is OK!);
Simplify your answer where possible and don't forget to add the constant of integration.

For the integral

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

we must first identify $u(x)$ and $\tfrac{dv}{dx}$ . In this case, let

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

Next, we need to calculate $\tfrac{du}{dx}$ and $v(x)$ :

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

Plugging these 4 terms into the integration by parts formula:

$\begin{split} \simplify{int(x^{n}ln({a}x),x)} &\,= \simplify[fractionNumbers]{1/{n+1} ln({a}x)x^{n+1}} - \simplify[fractionNumbers,alwaysTimes]{int((1/x) * (x^{n+1}/{n+1}),x)}, \\ \\ &\,= \simplify[fractionNumbers]{1/{n+1} ln({a}x) x^{n+1} - 1/{n+1} int(x^{n},x)}, \\ \\ &\,=\simplify[fractionNumbers]{1/{n+1}ln({a}x)x^{n+1} - 1/{(n+1)^2}x^{n+1}}+c. \end{split}$

Finally, simplifying our solution:

$\simplify{int(x^{n}ln({a}x),x)} = \simplify[fractionNumbers]{1/{(n+1)^2}x^{n+1}*[{n+1}ln({a}x)-1]}+c.$

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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.
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Ben McGovern 2 years, 7 months ago

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Integration: By Parts 3

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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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Generated value: integer

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Generated value: `integer`