Problem From Heck

Problem-From-Heck-5.2

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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10 Comments.

  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. 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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  4. 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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