## Problem From Heck

Posted by mathfail
on June 30, 2012
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!

Source: http://calculushumor.weebly.com/3/post/2012/05/problem-from-heck-5.html

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

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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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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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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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The new link is found at http://www.calculushumor.com/3/post/2012/05/problem-from-heck-5.html

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