Luis's copy of Integration by parts

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Luis Hernandez 6 years, 4 months ago

Created this as a copy of Integration by parts.

Name	Status	Author	Last Modified
Integration by parts	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Luis's copy of Integration by parts	draft	Luis Hernandez	30/11/2018 18:34

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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	-7
b	integer	5
m	integer	3

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

-1

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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

Part a

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:

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

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Luis Hernandez 6 years, 4 months ago

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Generated value: `integer`

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Luis's copy of Integration by parts

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Luis Hernandez 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`