Integration 1 - Substitution - Numbas at mathcentre.ac.uk

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Integration 1 - Substitution.

Name	Status	Author	Last Modified
Indefinite integral by substitution	draft	Bill Foster	06/08/2020 08:41
Julie's copy of Indefinite integral by substitution	draft	Julie Crowley	06/02/2017 15:35
Integration 1 - Substitution	Ready to use	Katie Lester	01/06/2016 09:47
Integration 1 - Substitution (with limits)	Ready to use	Katie Lester	01/06/2016 09:47
Katie's copy of Integration 1 - Substitution	Ready to use	Katie Dexter	01/06/2016 09:47
cormac's copy of Julie's copy of Indefinite integral by substitution	draft	cormac breen	23/02/2017 18:29
MA1002 Integration by Substitution	Ready to use	Kieran Mulchrone	13/03/2019 07:44
Integration by substitution Exam Q3 2018	draft	Violeta CIT	16/04/2018 17:21
Indefinite integral by substitution (custom feedback)	draft	Clodagh Carroll	29/01/2020 16:44
Blathnaid's copy of cormac's copy of Julie's copy of Indefinite integral by substitution	draft	Blathnaid Sheridan	16/04/2021 12:45
Integration by Substitution	draft	Xiaodan Leng	10/07/2019 22:17
Integration 1 - Substitution	draft	Xiaodan Leng	11/07/2019 06:48

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Question statement

Integrate the following by substitution.

Don't forget the constant of integration ( $C$ ).

Give any introductory information the student needs.

			Name	Type	Generated Value

a	list	[ 4, 5, 5, 5, 3 ]
b	list	[ 7, -8, -6, -6, -9 ]
m	list	[ 9, 4, 5, 4, 9 ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
repeat(random(1..5),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: `list`

[ 4, 5, 5, 5, 3 ]

→ Used by:

Advice
"Unnamed part" - prompt
"Unnamed part" → "Gap 0." - Correct answer
"Unnamed part" → "Gap 1." - Correct answer
"Unnamed part" → "Gap 2." - Correct answer
"Unnamed part" → "Gap 3." - Minimum accepted value
"Unnamed part" → "Gap 3." - Maximum accepted value

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
3. Gap Gap 2. Mathematical expression
  Add
4. Gap Gap 3. Number entry
  Add
5. Gap Gap 4. Mathematical expression
  Add

Gap-fill

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

$I=\simplify[std]{Int( x*({a[0]}x^2+{b[0]})^{m[0]},x)}$

Use $u=\simplify[std]{{a[0]}x^2+{b[0]}}$ as your substitution.

$\frac{du}{dx}=$ Gap 1.

$dx=$ Gap 2.

Substituting back into the original equation for $dx$ and pulling out constants gives

$I=$ Gap 3. $\simplify[std]{Int(u^{m[0]},u)}$

The next step is to integrate.

$\simplify{Int(u^{m[0]},u)}=$ Gap 4.

Putting all of these results together, we get the final answer of:

Gap 0.

Advice

This problem is best solved by using substitution.
Note that if we let $u=\simplify[std]{{a[0]} * (x ^ 2) + {b[0]}}$ then $du=\simplify[std]{(2*{a[0]} * x)*dx }$
Hence we can replace $xdx$ by $\frac{1}{2*\var{a[0]}}du$ .

Hence the integral becomes:

$\begin{eqnarray*} I&=&\simplify[std]{Int((1/(2{a[0]}))u^{m[0]},u)}\\ &=&\simplify[all]{(1/(2{a[0]}))u^{m[0]+1}/{m[0]+1}+C}\\ &=& \simplify[all]{({a[0]} * (x ^ 2) + {b[0]})^{m[0]+1}/(2{a[0]}*({m[0]}+1))+C} \end{eqnarray*}$

A Useful Result
This example can be generalised.
Suppose

$I = \int\; f'(x)g(f(x))\;dx$
The using the substitution

$u=f(x)$ we find that

$du=f'(x)\;dx$ and so using the same method as above:

$I = \int g(u)\;du$
And if we can find this simpler integral in terms of

$u$ we can replace

$u$ by

$f(x)$ and get the result in terms of

$x$ .

Give a worked solution to the whole question.

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Part	Test	Passed?
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Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

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Integration 1 - Substitution

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Taxonomy: mathcentre

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Taxonomy: Context

Contributors

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Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `list`

Parts

Gap-fill

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Error

Integration 1 - Substitution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`