Does the repeated integral of $\ln x$ have a pattern?

I guess @GrumpyParsnip is right, I should refrain from answering questions in the comments.

Here is what i wrote in the comment.

Yes, it has a pattern. I brightly remember reading that on some Wikipedia page, but for the life of me, I can't find the source (hence i write this as a comment). So here's the answer: Define $$f_n(x)=\frac{x^n}{n!}(\ln x- H_n).$$ Then by differentiation we find that $$\frac{d}{dx} f_n(x)=f_{n-1}(x)$$, and since $f_0(x)=\ln x$, this answers your question.