Integration: By Substitution 1a i

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Christian Lawson-Perfect 2 years, 6 months ago

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Ben McGovern 4 years ago

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Ben McGovern 4 years, 8 months ago

Created this as a copy of Integration: Using a Table of Integrals 6.

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Question statement

Calculate

$\simplify{int({2a}x({a}x^2+{b})^{n},x)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

			Name	Type	Generated Value

a	integer	4
b	integer	-5
n	integer	2

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xxxxxxxxxx
 
random(2..5)

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Generated value: `integer`

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Advice

Since this integral is of the form

$\int g'(x)f(g(x))\,dx,$ we can use the method of substitution to calculate 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

$\simplify{int({2a}x({a}x^2+{b})^{n},x)},$

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

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 now rewrite the original integral in terms of $u$ :

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

(Note: It is important to see that both the function we are integrating, and the variable we are integrating with respect to, has changed.)

$\simplify[fractionNumbers]{int(u^{n},u) = u^{n+1}/{n+1} + c}.$

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

$\simplify[fractionNumbers]{u^{n+1}/{n+1} + c = ({a}x^2+{b})^{n+1}/{n+1} + c}.$

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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 1a i

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

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`