$ \int_0^1(\ln\left(\frac{1}{x}\right))^n $ dx

Question

find values of n for which the integral converges

$ \int_0^1 (\ln\left(\frac{1}{x}\right))^n $ dx
i am able to understand that Integral needs to be broken in two parts so that convergence can be checked at x=1 and x=0 separately.

for the first part if n<0 then its proper integral similarly for the second half n>0 then its proper integral

i need to check for the other remaining cases but unable to get how?

Answer 1

$\int _0^1(\ln\frac{1}{x})^n\mathrm{d}x\overset{t=\frac{1}{x}}{=}\int_1^{+\infty}(\ln t)^nt^{-2}\mathrm{d}t\overset{z=\ln t}{=}\int_0^{+\infty}z^n\mathrm{e}^{-z}\mathrm{d}z=\Gamma(n+1)=n!$