Integration: By Substitution 2b

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Christian Lawson-Perfect 2 years, 6 months ago

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Ben McGovern 4 years, 8 months ago

Created this as a copy of Integration: By Substitution 1b.

Name	Status	Author	Last Modified
Differentiation: Using a Table of Derivatives 2	Ready to use	Ben McGovern	26/09/2022 12:10
Differentiation: Using a Table of Derivatives 3	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 4	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 5a	Ready to use	Ben McGovern	26/09/2022 12:11
Differentiation: Using a Table of Derivatives 5b	Ready to use	Ben McGovern	26/09/2022 12:12
Differentiation: Using a Table of Derivatives 5c	Ready to use	Ben McGovern	26/09/2022 12:12
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Integration: Using a Table of Integrals 3b	Ready to use	Ben McGovern	27/09/2022 09:01
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Integration: Using a Table of Integrals 5a	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 5b	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: Using a Table of Integrals 5c	Ready to use	Ben McGovern	27/09/2022 09:01
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Integration: By Substitution 1b	Ready to use	Ben McGovern	23/02/2023 12:45
Integration: By Substitution 1c	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2a	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2b	Ready to use	Ben McGovern	27/09/2022 09:01
Integration: By Substitution 2c	Ready to use	Ben McGovern	27/09/2022 09:01
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There are 151 other versions that do you not have access to.

Question statement

Calculate

$\simplify{int({2*n*a}x*e^({a}x^2),x)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

			Name	Type	Generated Value

			a	integer	3
			n	integer	2

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Advice

Since this integral is of the form

$\int a \,g'(x)f(g(x))\,dx,$ where

$a$ is a constant, we can use the method of substitution to solve it.

Firstly, we must make a change of variables from $x$ to $u$ , where $u$ is equal to the 'inner' function, $g(x)$ .

So, for

$\simplify{int({2*n*a}x*e^({a}x^2),x)},$

let $u=\color{red}{\simplify{{a}x^2}}.$

Now, we need to calculate the differential, $du$ , where

$du = \left(\frac{du}{dx}\right)dx.$

Differentiating $u$ with respect to $x$ :

$\frac{du}{dx}= \simplify{{2a}x}.$

Therefore,

$\color{blue}{du = \simplify{{2a}x}\, dx}.$

We can see that rather than having the exact expression for the differential in the integral, we have a multiple of it instead. It is good practice to write the integral again so that it does include the differential, and the multiple on the outside. This can make it easier when expressing it in terms of $u$ :

$\simplify{int({2*n*a}x*e^({a}x^2),x)} =\var{n}\int \color{blue}{\simplify{{2a}x}}e^{\color{red}{\simplify{{a}x^2}}} \color{blue}{\text{d}x} = \var{n}\int e^\color{red}{u}\color{blue}{\text{d}u}.$

Calculating the integral in terms of $u$ :

$\simplify[fractionNumbers]{{n}int(e^u,u) = {n}e^u + c}.$

Finally, we must rewrite our solution back in terms of the original variable $x$ :

$\simplify[fractionNumbers]{{n}e^u + c = {n}e^({a}x^2) + c}.$

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This question is used in the following exams:

Integration by Substitution by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.
Integration by substitution by Jinhua Mathias in Maths 3-2020.

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Christian Lawson-Perfect 2 years, 6 months ago

Ben McGovern 4 years ago

Ben McGovern 4 years, 8 months ago

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Integration: By Substitution 2b

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 2 years, 6 months ago

Ben McGovern 4 years ago

Ben McGovern 4 years, 8 months ago

Error in variable testing condition

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Warning

Generated value: integer

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Generated value: `integer`