where $ is the integral symbol , how can i get a general solution to
the indefinite integral $ dx / ln(x) .
problem relates to the primes.
$\displaystyle \int_a^{b}\frac{1}{\ln(x)}dx$
Let $\displaystyle ln(x)=u\Rightarrow{x=e^{u}}$
Then $\displaystyle dx=e^{u}$
new limits of integration are ln(a) and ln(b)
So we have $\displaystyle \int_{\ln(a)}^{\ln(b)}\frac{e^{u}}{u}du$
this is the exponential integral and has the power series solution
$\displaystyle \sum_{n=0}^{\infty}\frac{x^n}{n\cdot{n!}}$
thanks M28 , your reply was helpful as it does form part of a solution
as far as mathcad is concerned .there does seem to be more to it which i do
not understand yet , involving Ei , the exponential integral .I wonder if captain black can explain this ?
Sloppy work on my part I of course meant the logarithmic integral:
$\displaystyle {\rm{li}}(x)=\int_0^x \frac{1}{\ln(t)}~dt$
or the offset logarithmic integral (to aviod the singularity at 1):
$\displaystyle
{\rm{Li}}(x)=\int_2^x \frac{1}{\ln(t)}~dt
$
But also the exponential integral:
$\displaystyle \int_1^x \frac{1}{\ln(t)}~dt=\int_0^{\ln(x)}\frac{e^u}{u}~d u={\rm{Ei}}(\ln(x))-{\rm{Ei}}(0)$
Now this has not been as careful as I would like so still probably needs checking some more.
See the Wikipedia artice on the logarithmic integral for more information.
RonL
I would like to show symbols . As i am very newbie , i'm in need of tutorials so as to better illustrate my questions .
I came across the integral when reading about Guass . It looked harmless untill i tried to integrate . Eventually i gave up and solved it symbolic using mathcad .Then i realised i was in over my head with the Ei.
My next quest is to understand how Ei is derived .
many thanks .
Exponential Integral -- from Wolfram MathWorld
It isnt derived it comes from the incapability to get a closed solution to $\displaystyle \int\frac{e^{x}}{x}dx$