TITLE: Solve $\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$ without complex integration.
QUESTION [13 upvotes]: Solve $$\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$$ without complex integration.
This integral can be very easily solved with contour integration, but how can you solve it without taking a tour in the complex plane?

REPLY [4 votes]: You may prove through differential equations that $\frac{1}{\cosh x}$ is almost a fixed point of the Fourier transform, meaning that
$$ \mathscr{F}\left(\frac{1}{\cosh x}\right) = \sqrt{\frac{\pi}{2}}\,\frac{1}{\cosh\frac{\pi s}{2}}$$
according to Mathematica's standard choice of normalization constants. Similarly, in the sense of distributions
$$ \mathscr{F}\left(\cos x\right) = \sqrt{\frac{\pi}{2}}\,\left[\delta(s-1)+\delta(s+1)\right]$$
hence, by parity, it follows that
$$ \int_{0}^{+\infty}\frac{\cos x}{\cosh x}\,dx = \frac{1}{2}\int_{\mathbb{R}}\frac{\cos x}{\cosh x}\,dx = \frac{\pi}{2}\cdot\left.\frac{1}{\cosh\frac{\pi s}{2}}\right|_{s=1} = \frac{\pi}{2\cosh\frac{\pi}{2}} = \frac{\pi e^{\pi/2}}{e^\pi-1}.$$