cos-x-1-x-2-dx-pi-e-

Question Number 80925 by M±th+et£s last updated on 08/Feb/20

\int_{- \infty}^{\infty} \frac{c o s (x)}{1 + x^{2}} d x = \frac{π}{e}

Commented by mathmax by abdo last updated on 10/Feb/20

I = \int_{- \infty}^{+ \infty} \frac{c o s (x)}{1 + x^{2}} d x \Rightarrow I = R e (\int_{- \infty}^{+ \infty} \frac{e^{i x}}{x^{2} + 1} d x) l e t

φ (z) = \frac{e^{i z}}{z^{2} + 1} \Rightarrow φ (z) = \frac{e^{i z}}{(z - i) (z + i)} r e s i d u s t h e o r e m g i v e

\int_{- \infty}^{+ \infty} φ (z d z ? = 2 i π R e s (φ, i) = 2 i π \frac{e^{- 1}}{2 i} = \frac{π}{e} \Rightarrow I = \frac{π}{e}