Indefinite integral by substitution

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Bill Foster

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Bill Foster 55 years, 2 months ago

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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

Find the following integral.

Input the constant of integration as $C$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	3
s1	integer	1
b	integer	1
m	integer	9

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Description

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

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Generated value: `integer`

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Parts

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1. Step Information only

Gap-fill

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

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

$I=\;$ Unnamed gap

Input numbers in your answer as integers or fractions and not as decimals.

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Advice

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

Hence the integral becomes:

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

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

$x$ .

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Bill Foster 55 years, 2 months ago

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Generated value: `integer`

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Indefinite integral by substitution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`