Indefinite integral by substitution

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Bill Foster

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Bill Foster 55 years, 3 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 15:34
Indefinite integral by substitution 3 (custom feedback)	draft	Clodagh Carroll	26/03/2019 20:30

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Question statement

Find the following integral.

Input the constant of integration as $C$ .

Input all numbers as integers or fractions.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
c	number	31
b	integer	5
f	number	-16
s1	integer	-1
b1	integer	-3
test	number	471
c1	number	-5

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

$I=\;$ Unnamed gap

Input the constant of integration as $C$ .

Input all numbers as integers or fractions not as decimals.

Click on Show steps if you need help. You will lose 1 mark if you do so.

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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Bill Foster 55 years, 3 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, 3 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

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Generated value: `integer`