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

JamesPartial 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.

DavidHow 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.

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

PhilA more deatiled approach to what James said.

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

TobbznObviously, âˆ«(1+x^4)^(-1) dx = âˆ«Î£(-1)^n * x^(4n) dx

= Î£(-1)^n * x^(4n+1)/(4n+1)

And I’m perfectly satisfied with that answer. Fuck partial fractions.

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

dookLet a = sqrt(i), then

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

so

âˆ«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?

Calculus HumorWhat do you mean?

DavidThat, sir, is Wolfram|Alpha.

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