A calculus problem by Sal Gard

Evaluate integral from 0 to infinity p(x)-x/ln(x) dx. If it is in the form a-e^-b, find a^(2b). Note: p(x) is the prime counting function.

Moderator's edit :

$\large \int_0^\infty \dfrac{p(x) - x}{\ln x} \, dx$

Let $p(x)$ denote the prime counting function, that $p(x)$ denotes the function counting the number of prime numbers less than or equal to some real number $x$ .

If the integral above is equal to $a - e^{-b}$ , where $a$ and $b$ are integers, find $a^{2b}$ .