Integration by parts

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Christian Lawson-Perfect

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Christian Lawson-Perfect 10 years, 1 month ago

Gave some feedback: Ready to use

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Bill Foster 55 years, 4 months ago

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Name	Status	Author	Last Modified
Integration by parts	Ready to use	Bill Foster	01/06/2016 09:47
Integration by parts 3	draft	Violeta CIT	06/10/2016 16:44
Hina's copy of Integration by parts 3	draft	Hina Ahmed	31/10/2016 14:17
Maria's copy of Integration by parts	draft	Maria Aneiros	27/05/2019 07:25
Integration by parts	draft	Kevin Bohan	05/06/2019 11:30
Integration by parts	draft	Andrew McKinley	25/09/2019 11:49

There are 13 other versions that do you not have access to.

Question statement

Find the following indefinite integrals.

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$ where needed.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	1
c	integer	-4
b	integer	5
s3	integer	-1
s2	integer	-1
s1	integer	1
a1	integer	7
a2	integer	-8

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Value

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1

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Description

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Generated value: `integer`

→ Used by:

Advice
"Part a)" - prompt
"Part a)" → "Unnamed gap" - Correct answer
"Part b)" - prompt
"Part b)" → "Unnamed gap" - Correct answer

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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1. Step Information only
Part b) Gap-fill
1. Gap Mathematical expression
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Part c) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$\displaystyle \int \simplify[std]{{a}x*sin({b}x+{c})} dx = \phantom{{}}$ Unnamed gap

Input all numbers as fractions or integers and not decimals.

Input the constant of integration as $C$ .

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

Advice

a)

The formula for integrating by parts is

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

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

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

Hence,

$\begin{eqnarray} \int \simplify[std]{{a}x*sin({b}x+{c})} dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{(-{a}/{b})*x*cos({b}x+{c})} - \int \left( \simplify[std]{(-{a}/{b})*cos({b}x+{c})}\right) dx \\ &=& \simplify[std]{(-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}) + C} \end{eqnarray}$

b)

For this part we choose $u = \simplify[std]{{a}x}$ and $\frac{dv}{dx} = \simplify[std]{cos({b}x+{c})}$ .

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

Hence,

$\begin{eqnarray} \int \simplify[std]{{a}x*cos({b}x+{c})} dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{({a}/{b})*x*sin({b}x+{c})} - \int \left( \simplify[std]{({a}/{b})*sin({b}x+{c})}\right) dx \\ &=& \simplify[std]{({a}/{b})*x*sin({b}x+{c}) + ({a}/{b^2})*cos({b}x+{c}) + C} \end{eqnarray}$

c)

Using the results from Parts a and b, we have

$\begin{eqnarray*}I &=& \int \simplify[std]{{a1}x*sin({b}x+{c})} dx + \int \simplify[std]{{a2}x*cos({b}x+{c})} dx\\ &=& \simplify[std]{{a1}*((-{a}/{b})*x*cos({b}x+{c}) + ({a}/{b^2})*sin({b}x+{c}))+{a2}*(({a}/{b})*x*sin({b}x+{c}) +({a}/{b^2})*cos({b}x+{c}))+C}\\ &=&\simplify[std]{(-{a1}/{b})*x*cos({b}x+{c}) + ({a1}/{b^2})*sin({b}x+{c})+({a2}/{b})*x*sin({b}x+{c}) +({a2}/{b^2})*cos({b}x+{c}) + C}\\ &=&\simplify[std]{({a2}/{b}*x+{a1}/{b^2})*sin({b}x+{c})+({-a1}/{b}*x+{a2}/{b^2})*cos({b}x+{c})+C} \end{eqnarray*}$
Hence

$\displaystyle \simplify[std]{f(x) = {a2}/{b}*x+{a1}/{b^2}}$

$\displaystyle \simplify[std]{g(x) = {-a1}/{b}*x+{a2}/{b^2}}$

Give a worked solution to the whole question.

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Part	Test	Passed?
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This question is used in the following exams:

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Christian Lawson-Perfect 10 years, 1 month ago

Bill Foster 55 years, 4 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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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, 1 month ago

Bill Foster 55 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`