Integration by parts 2 - cos and sin - Numbas at mathcentre.ac.uk

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Xiaodan Leng 5 years, 9 months ago

Created this as a copy of Integration by parts 2 - cos and sin.

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 integral.

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	3
b	integer	-9
d	integer	-3
s2	integer	-1
s1	integer	-1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(2..5)

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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 Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. 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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Part d) Gap-fill
1. Gap Mathematical expression
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Gap-fill

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

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

The formula for integration by parts is

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

What is the most suitable choice for $u$ and $\frac{dv}{dx}$ ?

$u =\;$ Gap 0.

$\frac{dv}{dx} =\;$ Gap 1.

Advice

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,

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

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Integration by parts 2 - cos and sin

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Taxonomy: mathcentre

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Contributors

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Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

The formula for integrating by parts is

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Error

Integration by parts 2 - cos and sin

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

The formula for integrating by parts is

Generated value: `integer`