積分解説11

$$\begin{array}{rcl}{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}x}{({\pi }^2+x^2)\cosh x}& =& \frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}x}{(1+x^2)\cosh \pi x}(x\mapsto \pi x)\\ & =& {\int }_0^{\mathrm{\infty }}{\mathcal{F}}^{-1}[\frac{1}{\pi }\frac{1}{1+x^2}]\mathcal{F}[\frac{1}{\cosh \pi x}]\mathrm{d}x\\ & =& {\int }_0^{\mathrm{\infty }}\frac{\exp (-2\pi |x|)}{\cosh \pi x}\mathrm{d}x(\because \text{補題2})\\ & =& \frac{2}{\pi }{\int }_0^1\frac{x^2}{\frac{1}{x}+x}\frac{\mathrm{d}x}{x}(\exp (-\pi x)\mapsto x)\\ & =& \frac{2}{\pi }{\int }_0^1(1-\frac{1}{1+x^2})\mathrm{d}x\\ & =& \frac{2}{\pi }-\frac{1}{2}\end{array}$$