Integration by parts

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Christian Lawson-Perfect

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Christian Lawson-Perfect 9 years, 11 months 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
Integration by parts 2 - cos and sin	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 3 - exponential	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 4 - logarithm	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 2 - cos and sin with limits	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 3 - exponential with limits	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 4 - logarithm with limits	Ready to use	joshua boddy	01/06/2016 09:47
Integration by parts 2 - cos and sin with limits 19/10	Ready to use	Katie Dexter	01/06/2016 09:47
Jo-Ann's copy of Integration by parts - logarithm	draft	Jo-Ann Lyons	25/04/2019 12:33
Maria's copy of Jo-Ann's copy of Integration by parts - logarithm	draft	Maria Aneiros	27/05/2019 07:24
Integration by parts 2 - cos and sin	draft	Xiaodan Leng	11/07/2019 06:46

There are 32 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 $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	5
b	integer	7
d	integer	-3
s2	integer	-1
s1	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..5)

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `integer`

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
1. Step Information only
Part b) Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

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

$I=\displaystyle \int \simplify[std]{({a}x+{b})*cos({c}x+{d})} dx$

$I=\;$ Unnamed gap

Input all numbers as fractions or integers and not decimals.

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 $u = \simplify[std]{({a}x+{b})}$ and $\displaystyle \frac{dv}{dx} = \simplify[std]{cos({c}*x+{d})}$ .

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

Hence,

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

b)

For this part we choose $u = \simplify[std]{({a}x+{b})^2}$ and $\displaystyle \frac{dv}{dx} = \simplify[std]{sin({c}*x+{d})}$ .

So $\displaystyle \frac{du}{dx}=\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\displaystyle v = \simplify[std]{-(1/{c})*cos({c}*x+{d})}$ .

Hence,

$\begin{eqnarray*}I= \int \simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \int v \frac{du}{dx} dx \\ &=& \simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) + (1/{c})*Int({2*a}*({a}x+{b})*cos({c}*x+{d}),x)} \\ &=& \simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*Int(({a}x+{b})*cos({c}*x+{d}),x)}\dots (*) \end{eqnarray*}$

But in Part a we have aready worked out $\displaystyle \simplify[std]{Int(({a}x+{b})*cos({c}*x+{d}),x)=(({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d})}$

So on substituting this in equation (*) we find:

$\begin{eqnarray*}I&=& \simplify[std]{({-1}/{c})*({a}x+{b})^2*cos({c}*x+{d}) +({2*a}/{c})*((({a}x+{b})/{c})*sin({c}*x+{d}) +({a}/{c^2})*cos({c}*x+{d}))+C}\\ &=& \simplify[std]{-(({a}*x+{b})^2/{c})*cos({c}*x+{d})+(({2*a}({a}x+{b}))/{c^2})*sin({c}*x+{d})+({2*a^2}/{c^3})*cos({c}*x+{d})+C} \end{eqnarray*}$

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This question is used in the following exams:

First Exam by Hayley Bishop in Hayley's workspace.
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Maths Support: Integration by parts by Christian Lawson-Perfect in Maths Support Wiki.
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Christian Lawson-Perfect 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

Parts

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b)

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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 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

Generated value: `integer`