I came across this challenging integral:
Prove $$I:=\int_0^{\infty} \frac{\cos(x)}{\cosh(x)}dx=\frac{\pi}{2\cosh(\pi/2)}$$
My attempt
$$\int_0^{\infty} \frac{\cos(x)}{\cosh(x)}dx=\frac{\sin(x)}{\cosh(x)}\Bigg]_0^\infty+\int_0^\infty \frac{\sin(x)\sinh(x)}{\cosh^2(x)}$$ I don't think this is the right way. Do I need to take the real part of $e^{ix}$? How do I solve this integral?
Your help is much appreciated
