Primitives of the natural logarithm

All right, the idea of what I'm trying to do is find an easy way to generate the primitives of the natural logarithm without having to perform iteration after iteration of integration:

$\displaystyle P_{k+1}[f(x)]=\int P_k dx$

$\displaystyle P_0[ln{x}]=ln{x}$

$\displaystyle P_n[ln{x}]=?$

After performing a few integrations, I seem to have found a nice solution:

$\displaystyle P_n[ln{x}]=\frac{x^n}{n!} (ln(x)-H(n))$

Where H(n) is the Harmonic Series:

$\displaystyle H(n)=\sum_{k=0}^{n}{\frac{1}{k}}$

I'd appreciate it if anyone could verify or disprove my result.

If anyone would like to see my work, I'd gladly send them a .pdf of it.