Integral $\int \frac{dx}{x^4+1}$

Question

Find $$\int \frac{dx}{x^4+1}$$

I found a possible solution to this question here

However, I was wondering if there is a "nicer" solution, that would more understandable to a person who isn't advanced in complex numbers.

Thank you!

The link that you found addresses the definite integral (from $-\infty$ to $\infty$) and uses special techniques (residue calculus). — , Apr 01 '15 at 08:01

score 7 · Answer 1 · answered Apr 01 '15 at 07:32

7

All polynomials can be factors into linear and quadratic factors:

$$x^4 +1 = x^4 +2x^2 + 1 - 2x^2 = (x^2 + 1)^2 - 2x^2 = (x^2 + \sqrt 2 x +1)(x^2 - \sqrt 2 x +1)$$

Thus your integral can be found using partial fraction.

answered Apr 01 '15 at 07:32

1

I was just typing the same answer, but you were faster. (+2) – mickep Apr 01 '15 at 07:33
Just wondering how you can +2? Um...... @mickep – Apr 01 '15 at 07:34
1

Sorry, that was a small joke... – mickep Apr 01 '15 at 07:35
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@john with teamwork, yes we can. :) – David H Apr 01 '15 at 07:36

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