# Math Help - How to integrate this

1. ## How to integrate this

1/(x^5+1)

2. Originally Posted by CrazyAsian
1/(x^5+1)
This simple looking integral turned out to be a monster. See what people came up with here

3. Originally Posted by CrazyAsian
1/(x^5+1)
The major step is as follows:

You need to factorize the polynomial into irreducible polynomials in $\mathbb{R}[x]$.

The best way to do this is to factorize this in $\mathbb{C}[x]$ in linear factors.

We need to find all solutions to,
$x^5 = -1 = \cos (\pi+2\pi n) + i \sin (\pi+2\pi n)$
De Moiver's theorem,
$x = \cos \left( \frac{\pi}{5} + \frac{2\pi n}{5} \right) + i \sin \left( \frac{\pi}{5} + \frac{2\pi n}{5} \right) \mbox{ for }n=0,1,2,3,4$

But why you want to do that I have no idea. I guess that is why you are a CrazyAsian.