Integration by parts

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Christian Lawson-Perfect

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Christian Lawson-Perfect 10 years ago

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Bill Foster 55 years, 2 months ago

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Name	Status	Author	Last Modified
Integration by parts	Ready to use	Bill Foster	01/06/2016 09:47
Katie's copy of Integration by parts	Ready to use	Katie Dexter	01/06/2016 09:47
Integration by parts 1	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 1 with limits	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts - Ch 1	Ready to use	Katie Dexter	01/06/2016 09:47
Ch1: Integration by parts	Ready to use	Graham Wynn	01/06/2016 09:47

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Question statement

Find the following indefinite integral.

Give any introductory information the student needs.

			Name	Type	Generated Value

s3	integer	1
c	integer	1
b	integer	2
m	integer	2

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Description

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Generated value: `integer`

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Parts

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$I=\displaystyle \int \simplify[std]{x*({b}x+{c})^{m}} dx$
You are given that

$I=\simplify[std]{({b}x+{c})^{m+1}/{b^2*(m+1)*(m+2)}*g(x)+C}$
For a polynomial

$g(x)$ . You have to find

$g(x)$ .

$g(x)=\;$ Unnamed gap

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

Advice

The formula for integrating by parts is

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

We choose $u = x$ and $\displaystyle \frac{dv}{dx} = \simplify[std]{({b}*x+{c})^{m}}$ .

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

Hence,

$\begin{eqnarray*} \displaystyle \int \simplify[std]{x*({b}x+{c})^{m}} dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*b})*Int (({b}*x+{c})^{m+1}, x)} \\ &=& \simplify[std]{(x/{(m+1)*b})*({b}*x+{c})^{m+1} - (1/{(m+1)*(m+2)*b^2})*({b}*x+{c})^{m+2}+C} \\ &=&\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+2)}x - ({b}x+{c}))+C}\\ &=&\simplify[std]{({b}*x+{c})^{m+1}/{(m+1)*(m+2)*b^2}*({b*(m+1)}x - {c})+C} \end{eqnarray*}$
The solution is:

$\simplify[std]{g(x)={b*(m+1)}*x-{c}}$ .

Give a worked solution to the whole question.

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This question is used in the following exams:

Integration: Integration By Parts Practice by Bill Foster in Bill's workspace.
Bradford: Integration By Parts Practice by Bill Foster in Bill's workspace.
Maths Support: Integration by parts by Christian Lawson-Perfect in Maths Support Wiki.
mathcentre: Integration by parts by Newcastle University Mathematics and Statistics in mathcentre.
mathcentre: Diagnostic test - Indefinite integration by Bill Foster in Bill's workspace.
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Christian Lawson-Perfect 10 years ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Integration by parts

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`