Playing around with floor functions, I thought about using the maximum and minumum of functions to “merge” 2 asymptotic graphs which intersected each other infinitely. I then thought of the classical trigonometric Dirichlet Integral. Here is a graph of our goal integral. Please note the sign function:
$$\int_\Bbb R f(x)dx=\int_{-\infty}^0 \max\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx+ \int_0^\infty \min\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx = \int_0^\infty \min\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)+ \max\left(-\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx =\int_0^\infty \frac{\cos(x) \text{sgn}\left(\frac{\sin(x)}x – \frac{\cos(x)}x\right)}{2 x} +\frac{\sin(x) \text{sgn}\left(\frac {\cos(x)}x – \frac{\sin(x)}x\right)}{2 x} + \frac{\sin(x)}{2 x} + \frac{\cos(x)}{2 x}-\frac{\cos(x) \text{sgn}\left(\frac{\sin(x)}x + \frac{\cos(x)}x\right)}{2 x}+ \frac{\sin(x) \text{sgn}\left(-\frac {\cos(x)}x – \frac{\sin(x)}x\right)}{2 x} + \frac{\sin(x)}{2 x} – \frac{\cos(x)}{2 x} dx =\frac\pi 2+ \int_0^\infty \frac{\cos(x) \text{sgn}\left(\frac{\sin(x)}x – \frac{\cos(x)}x\right)}{2 x} +\frac{\sin(x) \text{sgn}\left(\frac {\cos(x)}x – \frac{\sin(x)}x\right)}{2 x} -\frac{\cos(x) \text{sgn}\left(\frac{\sin(x)}x + \frac{\cos(x)}x\right)}{2 x}+ \frac{\sin(x) \text{sgn}\left(-\frac {\cos(x)}x – \frac{\sin(x)}x\right)}{2 x} dx=\frac\pi 2+\frac12 \int_0^\infty\frac{|\cos(x)+\sin(x)|}{x}-\frac{|\cos(x)-\sin(x)|}{x}dx=\boxed{ \frac\pi 2+\frac1{\sqrt 2} \int_0^\infty\frac{\left|\sin\left(x+\frac\pi 4\right)\right|}{x}-\frac{\left|\sin\left(x-\frac\pi 4\right)\right|}{x}dx }=\frac{\pi}{2}+ \lim_{x\to \infty}\left(\frac12\text{Ci}(x) \left[\text {sgn}\,\sin\left(x + \frac \pi4\right) -\text{sgn}\,\cos \left(x + \frac{\pi}4\right)\right] + \frac12 \text{Si}(x) \left[\text {sgn}\,\sin\left(x + \frac{\pi}4\right) + \text{sgn}\,\cos \left(x + \frac{\pi}4\right)\right]\right)\mathop=^{???}\pi$$
Our integral uses trigonometric integrals. I need to verify this solution. Here is a graph of that final integrand. Here is a graph of the actual problem; this plot is also a visual representation of our constant:
We used the Dirichlet Integral in one step. Here is a verification of the maximum and minimum function terms. Can you please verify this solution or give another way to evaluate it? Desmos and Wolfram Alpha cannot find a numerical approximation for the problem. Please correct me and give feedback!
Similar results which should have no typos:
$$\int_\Bbb R \min \left(\frac{\sin(x)}{x},e^{-|x|}\right)dx=2$$
$$\int_\Bbb R \max\left(\frac{\sin(x)}{x},e^{-|x|}\right)dx=\pi$$
Final result using Silver Ratio $\delta_S$:
$$\int_\Bbb R f(x)dx=\int_{-\infty}^0 \max\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx+ \int_0^\infty \min\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx=\frac\pi 2+\coth^{-1}\left(\sqrt 2\right)=\frac\pi 2+\ln\left(\sqrt 2+1\right)=\frac\pi 2 +\ln(\delta_S)= 2.452169913814439644463931016619543751126745027949188…$$
Best Answer
I will concentrate on this part of your proof, the calculation of this integral (the other steps of your proof are correct):
$$\int_{0}^{\infty} \frac{\Big|\sin\left(x+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(x-\frac{\pi}{4}\right)\Big|}{x} dx$$
We will make use of the Lobachevsky integral formula:
Let $f(x)$ a $\pi$-periodic function (continous or integrable over its period). with
$\displaystyle f(x+\pi) = f(x)$ and $f(\pi-x) = f(x)$, for $0\leq x <\infty$. Then
$\displaystyle \int_{0}^{\infty} \frac{\sin x}{x} f(x) dx = \int_{0}^{\frac{\pi}{2}} f(x) dx. $
Now, for your integral: If you make the change of variable $\displaystyle x=\frac{w}{2}$
$\displaystyle I=\int_{0}^{\infty} \frac{\Big|\sin\left(x+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(x-\frac{\pi}{4}\right)\Big|}{x} dx = \int_{0}^{\infty} \frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{w} dw $
Multiplying and dividing by $\sin w$:
$$I= \int_{0}^{\infty} \frac{\sin w}{w} \frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{\sin w} dw$$
The function
$$f(w) = \frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{\sin w}$$
is $\pi$-periodic
By the Lobachevsky integral formula:
$$\displaystyle I=\int_{0}^{\infty} \frac{\sin w}{w} \frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{\sin w} dw = \int_{0}^{\frac{\pi}{2}}\frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{\sin w}dw $$
Note that for $w\in\left(0,\frac{\pi}{2}\right)$
$$\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big|= \sin\left(\frac{w}{2}+\frac{\pi}{4}\right)$$
$$\Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|= -\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)$$
Hence
$$\displaystyle\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big| = \sqrt{2}\sin\left(\frac{w}{2}\right)\displaystyle $$ Therefore
$$ I = \int_{0}^{\frac{\pi}{2}}\frac{\Big|\sin\left(\frac{w}{2}+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(\frac{w}{2}-\frac{\pi}{4}\right)\Big|}{\sin w}dw = \sqrt{2}\int_{0}^{\frac{\pi}{2}}\frac{\sin\left(\frac{w}{2}\right)}{\sin w}dw$$
Using $\displaystyle \frac{\sin\left(\frac{w}{2}\right)}{\sin w} = \frac{1}{2}\sec\left(\frac{w}{2}\right)$
we have
$$I = \sqrt{2}\int_{0}^{\frac{\pi}{2}}\frac{\sin\left(\frac{w}{2}\right)}{\sin w}dw = \frac{\sqrt{2}}{2}\int_{0}^{\frac{\pi}{2}} \sec\left(\frac{w}{2} \right)dw = \sqrt{2} \operatorname{arctanh}\left(\frac{1}{\sqrt{2}}\right)$$
We can conclude
$$\boxed{\int_{0}^{\infty} \frac{\Big|\sin\left(x+\frac{\pi}{4}\right)\Big| - \Big|\sin\left(x-\frac{\pi}{4}\right)\Big|}{x} dx = \sqrt{2} \operatorname{arctanh}\left(\frac{1}{\sqrt{2}}\right)}$$
Therefore
$$\boxed{\int_{-\infty}^0 \max\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx+ \int_0^\infty \min\left(\frac{\cos(x)}{x},\frac{\sin(x)}{x}\right)dx = \frac{\pi}{2} +\operatorname{arctanh}\left(\frac{1}{\sqrt{2}}\right)} $$
Some numerical approximations from Wolfram:
The sum is 2.44206
while
$$\frac{\pi}{2} + \operatorname{arctanh}\left(\frac{1}{\sqrt{2}}\right)\approx 2.45216..$$