calculate-5-e-ix-3-e-ix-1-x-2-dx-

Question Number 37304 by math khazana by abdo last updated on 11/Jun/18

c a l c u l a t e \int_{- \infty}^{+ \infty} \frac{5 + e^{i x}}{(3 + e^{i x}) (1 + x^{2})} d x .

Commented by prof Abdo imad last updated on 16/Jun/18

l e t φ (z) = \frac{5 + e^{i z}}{(e^{i z} + 3) (z^{2} + 1)} t h e p o l e s o f φ a r e

i a n d - i s o

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i) b u t

φ (z) = \frac{5 + e^{i z}}{(e^{i z} + 3) (z - i) (z + i)} \Rightarrow

R e s (φ, i) = \frac{5 + e^{- 1}}{(e^{- 1} + 3) (2 i)}

\int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{5 + e^{- 1}}{2 i (3 + e^{- 1})} = π \frac{5 e + 1}{3 e + 1}

I = π \frac{5 e + 1}{3 e + 1} .