For $a>1, b>1;$the value of $\underset{0}{\overset{\infty }{\int }}\left(a^{-x}-b^{-x}\right)dx=$

$\underset{0}{\overset{\infty }{\int }}\left(a^{-x}-b^{-x}\right)dx$

$={\left(\frac{-1}{a^x \log a}+\frac{1}{b^x \log b}\right)}_0^{\infty }$

$=\left(\frac{(-1)}{\log a}\left(0\right)+\frac{1}{\log b}\left(0\right)\right)-\left(\frac{-1}{\log a}+\frac{1}{\log b}\right)$

$\left(\begin{array}{l}\therefore x\to \infty \Rightarrow a^x\to \infty \\ \Rightarrow \frac{{\displaystyle 1}}{{\displaystyle a^x}}\to 0(\therefore a>1)\\ Similarly\frac{{\displaystyle 1}}{{\displaystyle b^x}}\to 0\end{array}\right)$

$=\frac{1}{\log a}-\frac{1}{\log b}$