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How do you do partial fraction on $\frac2{x^2 - 8}$ ? Or are there other method of doing? I tried trig substitution but could not get the answer.

Mahdi Khosravi
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sage
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    Do you mean $(2/x^2) - 8$, or $2/(x^2 - 8)$? Also, please explain what you've tried so that people can give salient help. –  Aug 04 '13 at 07:40

3 Answers3

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$$\frac{2}{x^2-8}=\frac{2}{x^2-(2\sqrt2)^2}=\frac{2}{(x-2\sqrt2)(x+2\sqrt2)}=\frac{A}{x-2\sqrt2}+\frac{B}{x+2\sqrt2}$$

Adi Dani
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By using Maple, you can find the fractions in @Adi's answer easily.

 [> evalf(convert(2/(x^2-8),parfrac,x,real),2);

                            -.36/(x+2.8)+.36/(x-2.8)

The square factor $(x^2-8)$ does not factor over the integers.

Mikasa
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It's very simple:

$$\frac{2}{x^2-8} = \frac{2}{(x+2\sqrt 2)(x-2\sqrt 2)} = \frac{\frac{1}{4} \sqrt 2}{x-2\sqrt 2}-\frac{\frac{1}{4} \sqrt 2}{x+2\sqrt 2}$$

Thus, for example, $$\int \frac{2}{x^2-8} dx = \frac{1}{4} \sqrt 2 \int \frac{dx}{x-2\sqrt 2} - \frac{1}{4} \sqrt 2 \int \frac{dx}{x+2\sqrt 2}$$

Riccardo Del Monte
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  • Why repeat @Adi's earlier answer (except that you provide the numerics, which can be criticized, while Adi does not)? – Did Aug 04 '13 at 10:06
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    @Did Because, first, I wanted to show how to apply the algeabric process of partial fractions to an integral, as I think sage was asking (the tag in his question is "integration"). Also, in Adi's answer one numerator is $x^2$ instead of $2$, but I can't modify his post, since the site says I need at least 6 characters to modify. – Riccardo Del Monte Aug 04 '13 at 11:55
  • "to apply the algebr(a)ic process of partial fractions to an integral" This is exactly what the other answer does, no? // "one numerator is x2 instead of 2"... This is not so, where did you see a numerator equal to $x^2$? – Did Aug 04 '13 at 22:07
  • @Did Adi had already corrected the numerator before you commented – Riccardo Del Monte Aug 05 '13 at 08:31
  • And so what? Since this $x^2$ was obviously a typo (the next expression in @Adi's being correct from the start), I still fail to see the point. – Did Aug 05 '13 at 08:42