$\text{ The total number of distinct values of }\alpha \text{ such that }$${\int }_a^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}={\log }_e\left(\frac{9}{8}\right)\text{ is }$

${\int }_{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}$$={\int }_{\alpha }^{\alpha +1}\left[\frac{1}{x+\alpha }-\frac{1}{x+\alpha +1}\right]dx\text{ [Using partial fraction] }$

                                       $$\begin{gathered} ={\left.\log \left(\frac{(x+\alpha )}{(x+\alpha +1)}\right)\right|}_{\alpha }^{\alpha +1} \\ =\log \left(\frac{2\alpha +1}{2\alpha +2}·\frac{2\alpha +1}{2\alpha }\right) \end{gathered}$$

                                       $=\log \frac{9}{8}\text{ (Given) }$

$\text{⇒ }\frac{(2\alpha +1{)}^2}{\alpha (\alpha +1)}=\frac{9}{2}$

$\Rightarrow 8{\alpha }^2+8\alpha +2=9{\alpha }^2+9\alpha$

$\Rightarrow {\alpha }^2+\alpha -2=0\Rightarrow \alpha =1,-2$

$\text{ Total distinct values of }\alpha =2$