Indefinite integral of improper polynomial fraction

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Indefinite integral of improper polynomial fraction	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Luis's copy of Indefinite integral of improper polynomial fraction	draft	Luis Hernandez	30/11/2018 18:38

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Question statement

Integrate the following function $f(x)$

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

Remember that the correct integration of expressions such as $\displaystyle \frac{1}{x-a}$ is:

$\int \frac{1}{x-a}\;dx=\ln(|x-a|)+C$

Give any introductory information the student needs.

			Name	Type	Generated Value

p	integer	8
sp	integer	1
sn	integer	1
n	integer	1

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1. Gap Mathematical expression
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$\displaystyle{\int f(x)\;dx=\;\;}$ Unnamed gap

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

Input all numbers as fractions or integers and not decimals.

Advice

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

First, perform a 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:

$\displaystyle{\int f(x)\;dx=\simplify[std]{ (({ - n} * (x ^ 2)) / 2) + ({p} / 2) * ln(abs((1 + x) / (1 -x)))+C}}$

where we have used partial fractions to integrate $\displaystyle{\frac{1}{1-x^2}}$ .

Note carefully that in this example it is good practice to take the absolute value of the argument of $\ln$ .

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Indefinite integral of improper polynomial fraction

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Indefinite integral of improper polynomial fraction

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

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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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Generated value: `integer`