In [1], page 2, Edwards shows a tabulated table by Gauss, for $x$ (distinct and uniformly distributed values from $5\cdot 10^5$ to $3\cdot 10^6$), the count of primes$<x$, the symbol $\int \frac{dn}{\log n}$ and differences of previous computed quantities. In the following page the author discusses what might have understood Gauss by this amount $\int dn/\log n$. I know a standard topic in numerical analysis: Gauss quadratures. On page 2 is it also noted that Gauss knew the average of the density of prime numbers, and tabulations of primes that were published, thus I imagine that he computed using these informations the first column, this is the count of primes$<x$.
But I'm not sure what was the method for the second column, the symbol $\int \frac{dn}{\log n}$. For this reason I have the following question:
Question. How were these quantities $\int \frac{dn}{\log n}$ computed by Gauss? If you can give some detail, better.
References:
[1] Harold M. Edwards, Riemann's Zeta Function, Dover (edition) 2001.