dx-1-x-7-

Question Number 158492 by bobhans last updated on 05/Nov/21

\int \frac{dx}{1 + x^{7}} ?

Answered by Ar Brandon last updated on 05/Nov/21

x^{7} + 1 = 0 \Rightarrow x^{7} = e^{(2 k + 1) i π} \Rightarrow x_{k} = e^{(\frac{2 k + 1}{7}) i π}, k \in [0, 6] k \in Z

I = \int \frac{d x}{x^{7} + 1} = \int \frac{d x}{\underset{k = 0}{\prod^{6}} (x - x_{k})} = \int \underset{k = 0}{\sum^{6}} \frac{a_{k}}{x - x_{k}} d x, a_{k} = \frac{1}{7 x_{k}^{6}} = - \frac{x_{k}}{7}

I = - \frac{1}{7} \int \underset{k = 0}{\sum^{6}} \frac{x_{k}}{x - x_{k}} d x = - \frac{1}{7} \underset{k = 0}{\sum^{6}} x_{k} \ln ∣ x - x_{k} ∣ + C