About frullani integral 
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Frullani proof integrals 

Let $f:\left[ {0,\infty } \right] \to \mathbb R$ be a a continuous function such that $$
\mathop {\lim }\limits_{x \to0+ } f\left( x \right) = L
$$Prove that $$
\int\limits_0^{\infty}  {\frac{{f\left( {ax} \right) - f\left( {bx} \right)}}
{x}}dx $$ converges and calculate the value.
It is known that $\int_a^\infty (f(x)/x)\,\mathrm{d}x$ converges for all a>0, but nothing of $\lim\limits_{x\to\infty}f(x)$ is told.
Also, what if $a>b$ or $a<b$?

Let $a,b>0$, without loss of generality we assume $a<b$. Let $0<\varepsilon<R$. First we split up the integral:
$$\int_{\varepsilon}^R \frac{f(ax)-f(bx)}{x} \, dx = \int_{\varepsilon}^R \frac{f(ax)}{x} \, dx- \int_{\varepsilon}^R \frac{f(bx)}{x} \, dx$$ where $$ \int_{\varepsilon}^R \frac{f(ax)}{x} \, dx \stackrel{z:=a \cdot x}{=} \int_{a \cdot \varepsilon}^{a \cdot R} \frac{f(z)}{\frac{z}{a}} \cdot \frac{1}{a} \, dz = \int_{a \cdot \varepsilon}^{a \cdot R} \frac{f(z)}{z} \, dz$$ (similarily for the second integral), thus
$$\int_{\varepsilon}^R \frac{f(ax)-f(bx)}{x} \, dx = \underbrace{\int_{a \varepsilon}^{b \varepsilon} \frac{f(z)}{z} \, dz}_{=:I_1} -  \underbrace{\int_{a \cdot R}^{b \cdot R} \frac{f(z)}{z} \, dz}_{=:I_2}$$


*

*We have $$I_1 = \int_{a \varepsilon}^{b \varepsilon} \frac{f(z)}{z} \, dz \stackrel{y:= \frac{z}{\varepsilon}}{=} \int_a^b \frac{f(\varepsilon \cdot y)}{y} \, dy$$ 
Since $a,b>0$ (thus $[a,b] \ni y \mapsto \frac{1}{y} \in L^1([a,b])$) and $f(\varepsilon \cdot y) \to L$ as $\varepsilon \to 0$ for all $y \in [a,b]$ we can apply dominated convergence and obtain
$$I_1 \to \int_a^b \frac{L}{y} \, dy = L \cdot (\log b-\log a) \qquad (\varepsilon \to 0)$$

*We want to prove $I_2 \to 0$ as $R \to \infty$. Let $\delta>0$. Define $$I_R := \int_a^R \frac{f(z)}{z} \, dz$$
Since $\int_a^{\infty} \frac{f(z)}{z} \, dz$ converges by assumption, we know that $I_R$ is a cauchy-sequence, i.e. there exists $S_0$ such that for all $S,T \geq S_0$: $$|I_S-I_T| \leq \delta \tag{1}$$ Now choose $R_0>0$ such that $a \cdot R_0 \geq S_0$. Then we obtain from (1) for all $R \geq R_0$:
$$|I_2| = |I_{b \cdot R}-I_{a \cdot R}| \leq \delta$$
i.e. $I_2 \to 0$ as $R \to \infty$. 


Adding all up we obtain
$$\int_0^\infty \frac{f(ax)-f(bx)}{x} \, dx = L \cdot (\log b- \log a) = f(0) \cdot (\log b-\log a)$$
