Is the integral $$\int_1^\infty\frac{x^{-a} - x^{-b}}{\log(x)}\,dx$$ convergent, where $b>a>1$?
I think the answer lies in defining a double integral with $yx^{(-y-1)}$ and applying Tonelli's Theorem, but the integral of $\frac{x^{-a}}{\log x}$ is still not integrable. Any ideas?
If we make substitution $x=\log t$, then we get integral of the form $$ \int\limits_{0}^\infty\frac{e^{-(a-1)t}-e^{-(b-1)t}}{t}dt $$ Now result follows from Frullani's integral formula applied to the function $f(t) = e^{-t}$. Moreover, this formula gives us exact value of the integral $$ \int\limits_{0}^\infty\frac{e^{-(a-1)t}-e^{-(b-1)t}}{t}dt=(f(0)-f(\infty))\log\frac{b-1}{a-1}=\log\frac{b-1}{a-1} $$
As for the proof of this formula see this discussion.
