it seems you cant use the property ln x^n = n ln x. I'm thinking there's integration by parts involved but not sure.
ln x^{n} = t x = e^{t} dx = e^{t} dt so initial eqn becomes [tex]\int t^n e^t dt[/tex] and now integrate by parts
[tex]ln \ x^n = t[/tex] [tex]e^{ln \ x^n} = e^t[/tex] [tex]x^n=e^t[/tex] [tex]\frac{d}{dt} \ (x^n)=\frac{d}{dt} \ (e^t)[/tex] [tex]0=e^t[/tex] remember that: [tex]x=exp \ y \Leftrightarrow y=ln \ x[/tex] So [tex]0=e^t \Leftrightarrow t = ln \ 0[/tex] Since ln 0 is undefined, so t is undefined too...
The problem is that this formula is [itex]\ln(x^n)=n\ln(x)[/itex], but you are now interested in [itex](\ln(x))^n[/itex] which is different. You probably knew this, but didn't sound very sure about it. Well tiny-tim of course answered quite sufficiently already, but I thought I would like to say that personally I like writing recursive formulas such as this: [tex] (\ln(x))^n = D_x \big(x(\ln(x))^n\big) - n(\ln(x))^{n-1} [/tex] fundoo, optics.tech, those were quite confusing comments
Umm there's a difference between (ln(x))^2 and ln(x^2). The first is what you seem to have, the latter is 2*ln(x).