Let \(a>0\) and calculate \[ \int_0^\infty {\cos(x) \over (x^2 + a^2)^2}\, dx .\]
Solution
Start from the standard residue evaluation \[ I(a):=\int_0^\infty\frac{\cos x}{x^2+a^2}\,dx =\frac{\pi}{2a}e^{-a}, \qquad a>0. \] Differentiation under the integral sign is justified locally in \(a>0\) by an integrable majorant, and gives \[ I'(a) =-2a\int_0^\infty\frac{\cos x}{(x^2+a^2)^2}\,dx. \] On the other hand, \[ I'(a) =-\frac{\pi e^{-a}}{2a} -\frac{\pi e^{-a}}{2a^2} =-\frac{\pi e^{-a}(a+1)}{2a^2}. \] Hence \[ \boxed{ \int_0^\infty\frac{\cos x}{(x^2+a^2)^2}\,dx =\frac{\pi e^{-a}(a+1)}{4a^3}.} \]