## Problem From Heck

This is a weekly “Problem From Heck” that has driven several high school teachers crazy. Try if you dare!

Thanks to Gregory Tewksbury-Calculus Humor for this submission!

** Note: Some posts on Math-Fail are user-submitted and NOT verified by the admin of the site before publication. If you find this post to be distasteful, non-math related, ?or something worse?, then definitely leave a comment letting me know. Thanks very much! Mike **

1. Partial fractions turns it into the sum of two quadratics, complete the squares, and then use a trig substitution in each of the resulting integrals. Incredibly simple. Anyone who has trouble with this should not be teaching mathematics.

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• How can you do partial fractions on this? It is not possible to factor x^4+1 into the product of two quadratics with integer coefficients.

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2. This also should give credit to Kaelyn Willingham, who is a co-owner of Calculus Humor.

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3. A more deatiled approach to what James said.

http://www.wolframalpha.com/input/?i=integrate+%281%2F%28x%5E4%2B1%29%29dx

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4. Obviously, ∫(1+x^4)^(-1) dx = ∫Σ(-1)^n * x^(4n) dx
= Σ(-1)^n * x^(4n+1)/(4n+1)
And I’m perfectly satisfied with that answer. Fuck partial fractions.

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• you have some (uniform) convergence issues there and you should refer to Lebesgue’s integration theory

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5. Let a = sqrt(i), then

1/(x^4+1) = 1/4 [1/(x+ia) - 1/(x-ia) - i/(x+a) + i/(x-a)]

so

∫dx/(x^4+1) = 1/4 [log (x+ia) - log (x-ia) - i log(x+a) + i log(x-a)]

Insert something about branches of the logarithm…, simplify and collect terms to taste.

WTF was that 2 page clusterf**k at the link?

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• What do you mean?

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• That, sir, is Wolfram|Alpha.

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