dx-1-x-12-

Question Number 129645 by liberty last updated on 17/Jan/21

ϝ = \int \frac{d x}{1 + x^{12}} .

Answered by mathmax by abdo last updated on 17/Jan/21

roots of z^{12} + 1 = 0 \Rightarrow z^{12} = e^{i π + 2 ik π} = e^{i (2 k + 1) π} \Rightarrow z_{k} = e^{i \frac{2 k + 1) π}{12}} withk \in [[0, 11]]

and \frac{1}{z^{12} + 1} = \frac{1}{\prod_{k = 0}^{11} (z - z_{k})} = \sum_{k = 0}^{11} \frac{a_{k}}{z - z_{k}}

a_{k} = \frac{1}{{12 z}_{k}^{11}} = \frac{z_{k}}{- 12} \Rightarrow a_{k} = - \frac{1}{12} \sum_{k = 0}^{11} \frac{z_{k}}{z - z_{k}} \Rightarrow

F = - \frac{1}{12} \sum_{k = 0}^{11} z_{k} \int \frac{dx}{x - z_{k}} = - \frac{1}{12} \sum_{k = 0}^{11} z_{k} \ln (x - z_{k}) + C