How to look for $\int \frac{1}{\ln x}\ dx$? I wonder if this form of integral could ever have an analytical solution? If not, why?
I've tried substitution with $u=\ln x$. So, $du=\frac{1}{x}\ dx$ and the original integral is equivalent to $\int \frac{e^u}{u}du$. But I'm unable to proceed further from here. Hope somebody could help me!
On
Note that this is also called the logarithmic integral function denoted by $\operatorname {li}(x) $ and is defined by: $$\operatorname {li}(x) = \int_{0}^{x} \frac {\mathrm dt }{\ln t} $$
Unfortunately there is no closed form solution to this integral. All you can do is to solve it using Taylor series.