Integration by substitution

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Simon Vaughan

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Simon Vaughan 4 years, 7 months ago

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Integration by substitution	Ready to use	Newcastle University Mathematics and Statistics	30/07/2020 10:43
Vinitha's copy of Integration by substitution	draft	Vinitha Sebastine	09/06/2016 11:18
Hina's copy of Integration by substitution	draft	Hina Ahmed	31/10/2016 12:53
Hina's copy of Integration by substitution	draft	Hina Ahmed	09/11/2016 13:34
Patrick's copy of Integration by substitution	draft	Patrick Joyce	18/07/2017 16:06
Luis's copy of Integration by substitution	draft	Luis Hernandez	30/11/2018 18:27
Integration by substitution	Ready to use	Simon Vaughan	20/08/2020 13:01

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Question statement

Find the following integral.

You must input the constant of integration as $C$ .

Input all numbers as integers or fractions.

You can click on Show steps to get a hint. You will lose 1 mark if you do so.

Note that $\displaystyle \int \frac{1}{x}\;dx=\ln(|x|)+C$ and you must include the absolute value in the argument of $\ln$ . You input $|x|$ as abs(x).

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	7
b	integer	-3
s1	integer	-1

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Generated value: `integer`

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"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer
"Unnamed part" → "Unnamed step" - prompt

Parts

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1. Gap Mathematical expression
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Gap-fill

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

$I=\;$ Unnamed gap

Input all numbers as integers or fractions.

Do not forget to include the constant of integration $C$ .

Advice

This exercise is best solved by using substitution.

Note that the numerator $\simplify[std]{{2 * a} * x + {b}}$ of

$\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}$ is the derivative of the denominator

$\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$

So if you use as your substitution $u=\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$

Hence we can replace $\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$

Hence the integral becomes:

$\begin{eqnarray*} I&=&\int\;\frac{du}{u}\\ &=&\ln(|u|)+C\\ &=& \simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \end{eqnarray*}$

A Useful Result
This example can be generalised.

Suppose

$I = \int\; \frac{f'(x)}{f(x)}\;dx$

The using the substitution $u=f(x)$ we find that $du=f'(x)\;dx$ and so using the same method as above:

$I = \int \frac{du}{u} = \ln(|u|)+ C = \ln(|f(x)|)+C$

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Simon Vaughan 4 years, 7 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Integration by substitution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Simon Vaughan 4 years, 7 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`