Integration of $\frac{1}{\log(x)}$

Question

Please integrate $\frac{1}{\log(x)}$ and illustrate the steps of your method of integration. I have already tried integration by parts following the ILATE rule and otherwise. Eventually it forms a loop that takes me back to the direct or substituted term where I started the process of integration.

Answer 1

By definition the integral of $1/\log x$ is $\mathrm{li}(x)$ (the logarithmic integral), see here.

Answer 2

$\int\dfrac{1}{\log x}dx=\int\dfrac{\ln10}{\ln x}dx$

Let $u=\ln x$ ,

Then $x=e^u$

$dx=e^u~du$

$\therefore\int\dfrac{\ln10}{\ln x}dx$

$=\int\dfrac{e^u\ln10}{u}du$

$=\int\sum\limits_{n=0}^\infty\dfrac{u^n\ln10}{n!u}du$

$=\int\sum\limits_{n=0}^\infty\dfrac{u^{n-1}\ln10}{n!}du$

$=\int\left(\dfrac{\ln10}{u}+\sum\limits_{n=1}^\infty\dfrac{u^{n-1}\ln10}{n!}\right)du$

$=\ln10\ln u+\sum\limits_{n=1}^\infty\dfrac{u^n\ln10}{n!n}+C$

$=\ln10\ln\ln x+\sum\limits_{n=1}^\infty\dfrac{(\ln x)^n\ln10}{n!n}+C$