find-the-value-of-0-cos-x-1-x-2-dx-

Question Number 26360 by abdo imad last updated on 24/Dec/17

f i n d t h e v a l u e o f \int_{0}^{\propto} \frac{c o s (α x)}{1 + x^{2}} d x .

Commented by abdo imad last updated on 25/Dec/17

l e t p u t I = \int_{0}^{\infty} \frac{c o s (α x)}{1 + x^{2}} d x \Rightarrow 2 I = \int_{R} \frac{c o s (α x)}{1 + x^{2}} d x

= R e (\int_{R} \frac{e^{i α x}}{1 + x^{2}} d x) a n d b y r e s i d u s t h e r e m

\int_{R} f (z) d z = 2 i π R e s (f, i) / f (z) = \frac{e^{i α z}}{1 + z^{2}}

R e (f, i) = {l i m}_{z - > i} (z - i) f (z) = \frac{e^{- α}}{2 i}

\int_{R} f (z) d z = 2 i π \frac{e^{- α}}{2 i} = π \overset{- α}{e}

\Rightarrow \int_{0}^{\infty} \frac{c o s (α x)}{1 + x^{2}} d x = \frac{π}{2} e^{- α} . (l o o k t h a t I m (\int_{R} \frac{e^{i α x}}{1 + x^{2}} d x) = 0)