Real Analysis Methodologies to show $\gamma =2\int_0^\infty \frac{\cos(x^2)-\cos(x)}{x}\,dx$
For $$\Gamma '\left ( x \right )=\int_{0}^{\infty }e^{-t}t^{x-1}\ln t\, \mathrm{d}t$$ using $$\ln t=\int_{0}^{\infty }\frac{e^{-s}-e^{-ts}}{s}\, \mathrm{d}s$$ we have $$\Gamma '\left ( x \right )=\int_{0}^{\infty }e^{-t}t^{x-1}\int_{0}^{\infty }\frac{e^{-s}-e^{-ts}}{s}\, \mathrm{d}s\mathrm{d}t=\Gamma \left ( x \right )\int_{0}^{\infty }\left ( e^{-s}-\frac{1}{\left ( s+1 \right )^{x}} \right )\frac{\mathrm{d}s}{s}$$ Hence, let $x=1$ we get $$\gamma =\int_{0}^{\infty }\left ( \frac{1}{s+1 }-e^{-s} \right )\frac{\mathrm{d}s}{s}$$ let $s=t^k,~k>0$, we get $$\gamma =\int_{0}^{\infty }\left ( \frac{1}{t^{k}+1 }-e^{-t^{k}} \right )\frac{k\, \mathrm{d}t}{t}$$ So,let $k=a,b$ $$\frac{\gamma}{a} =\int_{0}^{\infty }\left ( \frac{1}{t^{a}+1 }-e^{-t^{a}} \right )\frac{ \mathrm{d}t}{t}~~,~~\frac{\gamma}{b} =\int_{0}^{\infty }\left ( \frac{1}{t^{b}+1 }-e^{-t^{b}} \right )\frac{ \mathrm{d}t}{t}$$ hence $$\frac{\gamma}{b}-\frac{\gamma}{a} =\int_{0}^{\infty }\left [\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )+\left ( e^{-x^a}-e^{-x^b} \right ) \right ]\frac{ \mathrm{d}t}{t}$$ then $$\int_{0}^{\infty }\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}=\int_{0}^{1}\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}+\int_{1}^{\infty }\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}$$ let $t\rightarrow \dfrac{1}{t}$,we get $$\int_{0}^{1}\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}=-\int_{1}^{\infty }\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}$$ So $$\int_{0}^{\infty }\left ( \frac{1}{t^{b}+1 }- \frac{1}{t^{a}+1 } \right )\frac{\mathrm{d}t}{t}=0$$ and $$\left ( \frac{1}{b}-\frac{1}{a} \right )\gamma =\int_{0}^{\infty }\frac{e^{-t^a}-e^{-t^b}}{t}\, \mathrm{d}t\tag1$$ Lemma:
$$\int_{0}^{\infty }\frac{e^{-t^a}-\cos t^a}{t}\, \mathrm{d}t=0~,~a>0$$
Proof: Let $$f\left ( x \right )=\int_{0}^{\infty }\frac{e^{-t}-\cos t}{t}\, e^{-xt}\, \mathrm{d}t$$ so $$f'\left ( x \right )=\int_{0}^{\infty }\left ( \cos t-e^{-t} \right )e^{-xt}\, \mathrm{d}t=\frac{x}{1+x^2}-\frac{1}{1+x}$$ hence $$\int_{0}^{\infty }f'\left ( x \right ) \mathrm{d}x=\ln\frac{\sqrt{1+x^2}}{1+x}\Bigg|_{0}^{\infty }=0=f\left ( \infty \right )-f\left ( 0 \right )$$ It's easy to see that $f\left ( \infty \right )=0$,so $$f\left ( 0 \right )=\int_{0}^{\infty }\frac{e^{-t}-\cos t}{t}\, \mathrm{d}t=0$$ Let $x^a\to t$, we get $$a\int_{0}^{\infty }\frac{e^{-t^{a}}-\cos t^{a}}{t}\, \mathrm{d}t=0\Rightarrow \int_{0}^{\infty }\frac{e^{-t^{a}}-\cos t^{a}}{t}\, \mathrm{d}t=0\tag2$$ Now using $(1)$ and $(2)$, we get $$\Large\boxed{\color{Blue} {\int_{0}^{\infty }\frac{\cos x^{a}-\cos x^b}{x}\, \mathrm{d}x=\left ( \frac{1}{b}-\frac{1}{a} \right )\gamma }}$$
It turns out that we have the following observation:
Observation. For a nice function $f : [0,\infty) \to \Bbb{C}$, we have
$$ \int_{\epsilon}^{\infty} \frac{f(x)}{x} \, dx = -f(0)\log\epsilon + c(f) + o(1) \qquad \text{as } \epsilon \to 0^+ \tag{1} $$
where the constant $c(f)$ is computed by
$$ c(f) = \lim_{R\to\infty}\left( \int_{0}^{R} \mathcal{L}f(s) \, ds - f(0)\log R\right) - f(0)\gamma. \tag{2} $$
The reasoning is surprisingly simple: First, define $g(x) = (f(x) - f(0)\mathbf{1}_{(0,1)}(x))/x$ and notice that
$$ \int_{\epsilon}^{\infty} \frac{f(x)}{x} \, dx = -f(0)\log\epsilon + \int_{\epsilon}^{\infty} g(x) \, dx. $$
Assuming that the LHS of $\text{(1)}$ exists for all $\epsilon > 0$ and that $f$ behaves nice near $x = 0$, this implies $\text{(1)}$. Next, notice that $c(f) = \mathcal{L}g(0)$ and that $-(\mathcal{L}g(s))' = \mathcal{L}f(s) - f(0) (1-e^{-s})/s$. Therefore
\begin{align*} c(f) &= \lim_{R\to\infty} \int_{0}^{R} (-\mathcal{L}g(s))' \, ds \\ &= \lim_{R\to\infty} \left( \int_{0}^{R} \mathcal{L}f(s) \, ds - f(0) (1 - e^{-R})\log R + f(0) \int_{0}^{R} e^{-s}\log s \, ds \right) \\ &= \lim_{R\to\infty} \left( \int_{0}^{R} \mathcal{L}f(s) \, ds - f(0) \log R \right) - f(0)\gamma. \end{align*}
At this moment this is just a heuristic computation. For a broad class of functions for which the LHS of $\text{(1)}$ exists, however, this computation can be made rigorous. This is particularly true for our function $f(x) = \cos x$. Now plugging $\mathcal{L}f(s) = \frac{s}{s^2+1}$ shows that $c(f) = -\gamma$ and thus
$$ \int_{\epsilon}^{\infty} \frac{\cos x}{x} \, dx = -\log\epsilon - \gamma + o(1). $$
Plugging this asymptotics, we have
$$ \int_{\epsilon}^{\infty} \frac{\cos(x^2) - \cos x}{x} \, dx = \frac{1}{2}\int_{\epsilon^2}^{\infty} \frac{\cos x}{x} \, dx - \int_{\epsilon}^{\infty} \frac{\cos x}{x} \, dx = \frac{1}{2}\gamma + o(1) $$
and the identity follows by letting $\epsilon \to 0^+$.
Here, the constant $c(f)$ can be thought as a regularized value of the divergent integral $\int_{0}^{\infty} \frac{f(x)}{x} \, dx$. This has the following nice properties (whenever they exist)
- $c$ is linear: $c(\alpha f(x) + \beta g(x)) = \alpha c(f) + \beta c(g)$.
- $c(f(x^p)) = \frac{1}{p}c(f)$ for $p > 0$,
- $c(f(px)) = c(f) - f(0)\log p$ for $p > 0$,
Together with some known values, we can easily compute other types of integrals. For instance, using the fact that $c(\cos x) = -\gamma$ and $c(e^{-x}) = -\gamma$, we have
\begin{align*} \int_{0}^{\infty} \frac{\cos (x^p) - \exp(-x^q)}{x} \, dx &= c\left\{ \cos (x^p) - \exp(-x^q) \right\} \\ &= \frac{1}{p}c(\cos x) - \frac{1}{q}c(e^{-x}) = \gamma\left( \frac{1}{q} - \frac{1}{p}\right) \end{align*}
for $p, q > 0$.
I thought it might be instructive to post a solution that leverages the Lemma posted by @Renascence_5. To that end, we proceed.
The Lemma proved in the aforementioned post is expressed as
$$\int_0^\infty \frac{e^{-x^a}-\cos(x^a)}{x}\,dx=0 \tag 1$$
for $a>0$.
We now examine a generalized version of the integral of interest and write
$$I(a,b)=\int_0^\infty \frac{\cos(x^a)-\cos(x^b)}{x}\,dx \tag 2$$
for $a>0$ and $b>0$.
Using $(1)$ reveals that $(2)$ can be written as
$$I(a,b)=\int_0^\infty \frac{e^{-x^a}-e^{-x^b}}{x}\,dx \tag 3$$
Next, we integrate by parts the integral in $(3)$ with $u=e^{-x^a}-e^{-x^b}$ and $v=\log(x)$ to obtain
$$\begin{align} I(a,b)&=\int_0^\infty \left(ax^{a-1}e^{-x^a}-bx^{b-1}e^{-x^b}\right)\,\log(x)\,dx\\\\ &=\int_0^\infty ax^{a-1}e^{-x^a}\,\log(x)\,dx-\int_0^\infty bx^{b-1}e^{-x^b}\,\log(x)\,dx\\\\ &=\frac1a \int_0^\infty e^{-x}\,\log(x)\,dx-\frac1b \int_0^\infty e^{-x}\,\log(x)\,dx\\\\ &=-\left(\frac1a -\frac1b\right)\,\gamma \end{align}$$
where we used the integral relationship $\gamma =-\int_0^\infty e^{-x}\,\log(x)$.
NOTE:
We can show that $\gamma$ as given by $\gamma=-\int_0^\infty e^{-x}\,\log(x)\,dx$ is equal to $\gamma$ as expressed by the limit $\gamma=\lim_{n\to \infty}\left(-\log(n)+\sum_{k=1}^n\frac1k\right)$ as follows.
$$\begin{align} \int_0^\infty e^{-x}\,\log(x)\,dx&=\lim_{n\to \infty}\int_0^n \left(1-\frac xn\right)^n\,\log(x)\,dx\\\\ &=\lim_{n\to \infty} n \int_0^1 x^n \log(n(1-x))\,dx\\\\ &=\lim_{n\to \infty} n \left(\log(n) \int_0^1 x^n\,dx+\int_0^1 x^n\,\log(1-x)\,dx\right)\\\\ &=\lim_{n\to \infty} \left(\frac{n}{n+1}\log(n)+\int_0^1 x^n\,\log(1-x)\,dx\right)\\\\ &=\lim_{n\to \infty} \left(\frac{n}{n+1}\log(n)-n\sum_{k=1}^\infty \frac1k \int_0^1 x^{n+k}\,dx\right)\\\\ &=\lim_{n\to \infty} \left(\frac{n}{n+1}\log(n)-n \sum_{k=1}^\infty \frac1{k(k+n+1)}\right)\\\\ &=\lim_{n\to \infty} \frac{n}{n+1}\left(\log(n)- \sum_{k=1}^\infty \left(\frac1k-\frac1{k+n+1}\right)\right)\\\\ &=\lim_{n\to \infty} \frac{n}{n+1}\left(\log(n)- \sum_{k=1}^{n+1} \frac1k\right)\\\\ &=\lim_{n\to \infty} \left(\log(n)- \sum_{k=1}^{n} \frac1k\right)\\\\ \end{align}$$
as was to be shown!