 Solution

Jordan lemma states that if $a>0$ and $g$ is analytic in the upper half-plane outside a fixed circle, withthen on the upper semicircle $C_R$,Indeed, $|e^{iaR{e}^{i\theta }}|=e^{-aR\sin \theta }$. Using symmetry and $\sin \theta \ge 2\theta /\pi$ on $0\le \theta \le \pi /2$ gives

The residue of $f$ at an isolated singularity $a$ is the coefficient of $(z-a{)}^{-1}$ in its Laurent series, equivalentlyIf $f(z)=g(z)/(z-a{)}^k$, the Taylor expansion of the analytic function $g$ shows that

For the real integral, apply contour integration in the upper half-plane toJordan's lemma removes the semicircular arc. The only enclosed singularity is the double pole at $z=i$, whose residue isThe residue theorem therefore givesIts real part vanishes because $x^3\cos x/(1+x^2{)}^2$ is an odd function, while its imaginary part is the requested integral. Hence