TITLE: Integrate $\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$
QUESTION [8 upvotes]: I can't solve the integral
$$\int_0^\infty \frac{e^{-x/\sqrt3}-e^{-x/\sqrt2}}{x}\,\mathrm{d}x$$
I tried it by using Beta and Gamma function and integration by parts. Please help me to solve it.

REPLY [8 votes]: \begin{align}
\int_0^{\infty}e^{-yx}\ \mathrm dx
&=\frac{1}{y}\\[9pt]
\int_b^a\int_0^{\infty}e^{-yx}\ \mathrm dx\ \mathrm dy
&=\int_b^a\frac{\mathrm dy}{y}\\[9pt]
\int_0^{\infty}\int_b^ae^{-xy}\ \mathrm dy\ \mathrm dx&=\ln a-\ln b\\[9pt]
\int_0^{\infty}\left[-\frac{e^{-xy}}{x}\right]_b^a\ \mathrm dx
&=\ln\left(\frac{a}{b}\right)\\[9pt]
\int_0^{\infty}\frac{e^{-ax}-e^{-bx}}{x}\ \mathrm dx
&=\ln\left(\frac{b}{a}\right)\\[9pt]
\end{align}

$$
\int_0^{\infty}\frac{e^{-x/\sqrt{3}}-e^{-x/\sqrt{2}}}{x}\ \mathrm dx
=\bbox[8pt,border:3px #FF69B4 solid]{\color{red}{\large\frac{1}{2}\ln\left(\frac{3}{2}\right)}}
$$