Primitive of Reciprocal of Logarithm of x

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {\ln x} = \ln \left({\ln x}\right) + \ln x + \sum_{k \mathop \ge 2}^n \frac {\left({\ln x}\right)^k} {k \times k!} + C$

Proof
From Primitive of $\dfrac {x^m} {\ln x}$:
 * $\displaystyle \int \frac {x^m \ \mathrm d x} {\ln x} = \ln \left({\ln x}\right) + \left({m + 1}\right) \ln x + \sum_{k \mathop \ge 2}^n \frac {\left({m + 1}\right)^k \left({\ln x}\right)^k} {k \times k!} + C$

The result follows by setting $m = 1$.

Also see

 * Primitive of $\dfrac 1 {\ln x}$ has no Solution in Elementary Functions