find-the-value-of-0-arctan-sin-pix-2-x-2-pi-2-dx-

Question Number 76353 by mathmax by abdo last updated on 26/Dec/19

f i n d t h e v a l u e o f \int_{0}^{\infty} \frac{a r c t a n (s i n (π x^{2}))}{x^{2} + π^{2}} d x

Commented by mathmax by abdo last updated on 29/Dec/19

l e t A = \int_{0}^{\infty} \frac{a r c t a n (s i n (π x^{2}))}{x^{2} + π^{2}} d x \Rightarrow 2 A = \int_{- \infty}^{+ \infty} \frac{a r c t a n (s i n (π x^{2}))}{x^{2} + π^{2}}

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

r e s i d u s t h e o r e m g i v e \int_{- \infty}^{+ \infty} W (z) d z = 2 i π R e s (W, i π)

= 2 i π \times \frac{∣ a r c t a n (s i n (π (- π^{2})) ∣}{2 i π} = a r c t a n (s i n (π^{3}))

\Rightarrow A = \frac{a r c t a n (s i n (π^{3}))}{2}