Find $\int\ln^{n} x$ . 
Find $\int\ln^{n} x$ .

My observation
$\int\ln^{1} x= x(\ln x- 1)+ constant$
$\int\ln^{2} x= x(\ln^{2} x- 2\ln x+ 2)+ constant$
$\int\ln^{3} x= x(\ln^{3} x- 3\ln^{2} x+ 6\ln x- 6)+ constant$
$\int\ln^{4} x= x(\ln^{4} x- 4\ln^{3} x+ 12\ln^{2} x- 24\ln x+ 24)+ constant$
$$\ddots$$
We have
$$\ddots$$
$\frac{{\rm d}}{{\rm d}\ln x} (\ln^{4} x- 4\ln^{3} x+ 12\ln^{2} x- 24\ln x+ 24)= 4(\ln^{3} x- 3\ln^{2} x+ 6\ln x- 6)$
$\frac{{\rm d}}{{\rm d}\ln x} (\ln^{3} x- 3\ln^{2} x+ 6\ln x- 6)= 3(\ln^{2} x- 2\ln x+ 2)$
$\frac{{\rm d}}{{\rm d}\ln x} (\ln^{2} x- 2\ln x+ 2)= 2(\ln x- 1)$
I used these to prep for my tests, thanks!

$$ \int \ln^{n} x dx = x \ln^{n}x - n\int\ln^{n-1}x dx + C $$
Please see:
https://www.youtube.com/watch?v=xE0Pp4I7PiA
