findf-a-arctan-cosx-x-2-a-2-dx-witha-gt-0-

Question Number 74350 by mathmax by abdo last updated on 22/Nov/19

f i n d f (a) = \int_{- \infty}^{+ \infty} \frac{a r c t a n (c o s x)}{x^{2} + a^{2}} d x w i t h a > 0

Commented by abdomathmax last updated on 23/Nov/19

c h a n g e m e n t x = a t g i v e f (a) = \int_{- \infty}^{+ \infty} \frac{a r c t a n (c o s (a t))}{a^{2} (t^{2} + 1)} a d t

\Rightarrow a f (a) = \int_{- \infty}^{+ \infty} \frac{a r c t a n (c o s (a t))}{t^{2} + 1} d t l e t

W (z) = \frac{a r c t a n (c o s (a z))}{z^{2} + 1} \Rightarrow W (z) = \frac{a r c r a n (c o s (a z))}{(z - i) (z + i)}

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

2 i π \times \frac{a r c t a n (c o s (a i))}{2 i} = π a r c r a n (c o s (i a))

c o s (i a) = c h (- a) = c h (a) = \frac{e^{a} + e^{- a}}{2} \Rightarrow

\int_{- \infty}^{+ \infty} W (z) d z = π a r c t a n (\frac{e^{a} + e^{- a}}{2}) = a f (a) \Rightarrow

f (a) = \frac{π}{a} a r c t a n (\frac{e^{a} + e^{- a}}{2})