calculate-the-exact-value-0-cos-x-x-2-1-dx-

Question Number 82816 by M±th+et£s last updated on 24/Feb/20

c a l c u l a t e t h e e x a c t v a l u e

\int_{0}^{\infty} \frac{c o s (x)}{x^{2} + 1} d x

Commented by msup trace by abdo last updated on 24/Feb/20

I = \int_{0}^{\infty} \frac{c o s x}{1 + x^{2}} d x \Rightarrow I = \frac{1}{2} \int_{- \infty}^{+ \infty} \frac{c o s x}{1 + x^{2}} d x

\Rightarrow 2 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 d 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 (φ, i)

= 2 i π \times \frac{e^{- 1}}{2 i} = π e^{- 1} \Rightarrow

I = \frac{π}{2} e^{- 1} \Rightarrow I = \frac{π}{2 e}

Commented by M±th+et£s last updated on 24/Feb/20

t h a n k y o u s i r

Commented by mathmax by abdo last updated on 24/Feb/20

y o u a r e [w e l c o m e