Prove this asymptotic integral

Again, something has gone wrong in the telling; what is $\nu$ supposed to be?

I also don't know what you mean by $\log^p(t)$. Is it $(\log t)^p$, log(t) in base b (more normally written $\log_b(t)$), or the pth iterated logarithm $\underbrace{\log(\log(...\log(}_{p\text{ times}} p(t)))$, or something else altogether?

Also, it would be better if your post was an addendum to the previous one, rather than starting a completely new thread.