Question-65332

Question Number 65332 by aliesam last updated on 28/Jul/19

Commented by mathmax by abdo last updated on 29/Jul/19

l e t s u p p o s e n i n t e r l e t d e c o m p o s e t h e f r a c t i o n

F (x) = \frac{x}{x^{n} + 1}

z^{n} + 1 = 0 \Rightarrow z^{n} = - 1 \Rightarrow r^{n} e^{i n θ} = e^{(2 k + 1) π} (z = {r e}^{i θ}) \Rightarrow r = 1 a n d θ

θ = \frac{(2 k + 1) π}{n} s o t h e r o o t s a r e Z_{k} = e^{i \frac{(2 k + 1) π}{n}} k \in [[0, n - 1]]

F (x) = \sum_{i = 0}^{n - 1} \frac{λ_{i}}{x - Z_{i}} w i t h λ_{i} = \frac{Z_{i}}{{n Z}_{i}^{n - 1}} = \frac{Z_{i}^{2}}{- n} \Rightarrow

F (x) = - \frac{1}{n} \sum_{i = 0}^{n - 1} \frac{Z_{i}^{2}}{x - Z_{i}} \Rightarrow \int F (x) d x = - \frac{1}{n} \sum_{i = 0}^{n - 1} Z_{i}^{2} \int \frac{d x}{x - Z_{i}}

r e s t t o d e t e r m i n e \int \frac{d x}{x - Z_{i}} b e c o n t i n u e d \dots