I have revised this interesting stackexchange question with solution, $\int_{2}^{x-2}\frac{t}{\log(t)}\frac{1}{\log(x-t)}\mathrm{dt} \tag{1}$ in the form of this approximation :
$\frac{x}{\log(x-2)}\int_{2}^{x/2}\frac{dt}{\log(t)} \tag{2}$
now the rational it seems behind using $\frac{x}{\log(x-2)}$ outside the LogIntegral seems to ensure a bound in one direction but I'd like to get tighter bounds in both directions.
The question is how do I 'tighten' up the bounds on this particular approximation (2)?
What are the strategies for going about this sort of thing?
To help get things along since posting the question a couple of days ago, I've considered a loose bound as a first attempt. Maybe somebody can improve or tighten the bounds by some other method.
Using $\frac{x}{\log(x-2)}$ (as suggested by @Jack D'Aurizio ) outside the Log Integral one bound can be established and looking at the plot of $\frac{x}{\log(x)}$ and $\frac{x}{\log(x-t)}$, $\frac{x}{\log(x/2)}$ ensures the other bound such that:
$\frac{x}{\log(x-2)}\int_{2}^{x/2}\frac{dt}{\log(t)}<\int_{2}^{x-2}\frac{t}{\log(t)}\frac{1}{\log(x-t)}\mathrm{dt}<\frac{x}{\log(x/2)}\int_{2}^{x-2}\frac{dt}{\log(t)}$