what's the integral of $(-\ln(x))^a$

Question

I'm trying to fornd the CDF of the multiplication of n uniform R.V. I found that the pdf I $\frac{(-\ln(z))^{n-1}}{(n-1)!}$ for $0<z<1$. To find the CDF, I'm trying to integrate this. What's the integral $$\int(-\ln(x))^a dx\quad? $$

product distribution of two uniform distribution, what about 3 or more

Answer 1

You can create a recursion formula with integration by parts, with $u=\ln^a{x}$, $dv=dx$, yielding $$\int\ln^a{x}dx=x\ln^a{x}-a\int\ln^{a-1}x dx$$