A calculus problem by Sal Gard

Calculus Level 4

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 \(a^{2b} \).


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