Logarithmic Integral and Primes

[Frogeyedpeas]

Hey guys, I was reading a brief article which described the logarithmic integral for approximating π(x)

in two ways:

∫x/log(x) dx

and

∫1/log(x) dx

I am aware that the second is the actual definition of li(x) but the top is used extremely frequently and upon trying out the top it matches pi(x) very closely so I'm not sure which is correct or if both are in the running for being defined as logarithmic integral (though mathematica says only the second is)

 

[Frogeyedpeas]

Anybody?

 

[Norwegian]

I don't understand what statement you are making about the top integral, but If you remove the integral sign and dx, you have an expression that according to the PNT grows like π(x).

Your second integral, taken from 2 to x, also known as li(x), is an even better approximation to π(x)