Integration by parts

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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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.

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			Name	Type	Generated Value

s3	integer	1
c	integer	7
b	integer	2
m	integer	6

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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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This question is used in the following exams:

Integration by parts by Newcastle University Mathematics and Statistics in Content created by Newcastle University.
Integration by parts by Henrik Skov Midtiby in SDS.
Luis's copy of Integration by parts by Luis Hernandez in MAT333.
Antiderivadas e integrales by Luis Hernandez in Antiderivadas e integrales directas.
Integration and area by Adrian Jannetta in Foundation Maths.
Ann's copy of Integration by parts by Ann Smith in Calculus.
Integration by parts by Ann Smith in Calculus.
Newcastle University Integration by Nicolas Sawyer in Nicolas's workspace.
Integration by parts by Adrian Jannetta in Foundation Maths.
Integration by parts by Wan Mekwi in Introduction to Calculus.
Integration by parts -section 8E by Kuldeep Singh in Kuldeep's workspace.
claire's copy of Integration by parts -section 8E by claire lines in claire's workspace.
Integration by parts ME0006 weekly by claire lines in claire's workspace.

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Generated value: `integer`

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Integration by parts

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`