Simon's copy of Integrate algebraic fraction

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Simon Thomas 6 years, 1 month ago

Created this as a copy of Integrate algebraic fraction.

Name	Status	Author	Last Modified
Integrate algebraic fraction	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Luis's copy of Integrate algebraic fraction	draft	Luis Hernandez	30/11/2018 18:32
Simon's copy of Integrate algebraic fraction	draft	Simon Thomas	20/03/2019 15:01

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Question statement

Integrate the following function $f(x)$

$f(x)=\simplify[std]{({n}x^3+{n}x+{p})/(1+x^2)}$

You may need to use the standard result $\int \frac{1}{1+x^2}dx = \arctan{x}+C$

Give any introductory information the student needs.

			Name	Type	Generated Value

p	integer	9
sp	integer	1
sn	integer	-1
n	integer	-3

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xxxxxxxxxx
 
sp*random(1..9)

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Generated value: `integer`

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Statement
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"Unnamed part" → "Unnamed gap" - Correct answer

Parts

Gap-fill
1. Gap Mathematical expression
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Gap-fill

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

$\displaystyle \int f(x)\;dx=\;\;$ Unnamed gap

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

Also input all numbers as fractions or integers and not as decimals.

Advice

Since the degree of the numerator of $f(x)$ is greater than the denominator, $f(x)$ is improper.

First, perform a polynomial long division, so that $f(x)$ can be rewritten in the form $\displaystyle{f(x)=\simplify[std]{{n}x+{p}/(1+x^2)}}$ .

Each term of this expression can then be integrated using standard functions (to within the arbitrary constant) to give:

$\displaystyle{\int f(x)\;dx=\simplify[std]{{n}x^2/2+{p}arctan(x)} +C}$

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This question is used in the following exam:

Integration introduction by Simon Thomas in Calculus.

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Simon's copy of Integrate algebraic fraction

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Taxonomy: mathcentre

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Simon Thomas 6 years, 1 month ago

Error in variable testing condition

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Warning

Generated value: `integer`

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Gap-fill

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Error

Simon's copy of Integrate algebraic fraction

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 1 month ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`