Integration 1 - Substitution (with limits) - Numbas at mathcentre.ac.uk

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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

There are 34 other versions that do you not have access to.

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	[ 5, 2, 1, 2, 3 ]
b	list	[ 7, 4, 5, -2, -1 ]
m	list	[ 7, 5, 8, 8, 7 ]

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.

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Generated value: `list`

[ 5, 2, 1, 2, 3 ]

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Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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3. Gap Gap 2. Mathematical expression
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4. Gap Gap 3. Number entry
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5. Gap Gap 4. Mathematical expression
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6. Gap Gap 5. Mathematical expression
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7. Gap Gap 6. Mathematical expression
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8. Gap Gap 7. Mathematical expression
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Gap-fill

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

$I=\int_{1}^{2}\simplify[std]{x*({a[0]}x^2+{b[0]})^{m[0]} dx}$

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

First we integrate without the limits.

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

$dx=$ Gap 2.

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

$I=$ Gap 3. $\int{\simplify[std]{u^{m[0]} du}}$

The next step is to integrate.

$\int{\simplify{u^{m[0]} du}}=$ Gap 4.

Putting all of these together, we get the result in terms of $x$ (remembering the constant of integration):

Gap 0.

Now put the limits into the calculation for $x$ . You may neglect $C$ in these boxes, as it will be cancelled in the subtraction.

$[$ Gap 5. $]-[$ Gap 6. $]$

$=$ Gap 7.

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:

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

If there are limits, we substitute in the upper limit for $x$ and then subtract the result of substituting in the lower limit for $x$ .

This cancels the constant of integration and leaves us with a definite number.

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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This question is used in the following exam:

Block 3, Unit 3: Integration by Subsitution by Paul Abel in Paul's workspace.

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Integration 1 - Substitution (with limits)

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Katie Lester 9 years, 6 months ago

Katie Lester 9 years, 6 months ago

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Integration 1 - Substitution (with limits)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

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Katie Lester 9 years, 6 months ago

Katie Lester 9 years, 6 months ago

Error in variable testing condition

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Generated value: list

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Generated value: `list`