3.75 (4 votes)
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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
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.
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.
This also should give credit to Kaelyn Willingham, who is a co-owner of Calculus Humor.
A more deatiled approach to what James said.
http://www.wolframalpha.com/input/?i=integrate+%281%2F%28x%5E4%2B1%29%29dx
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.
you have some (uniform) convergence issues there and you should refer to Lebesgue’s integration theory
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?
What do you mean?
That, sir, is Wolfram|Alpha.
The new link is found at http://www.calculushumor.com/3/post/2012/05/problem-from-heck-5.html