I have to analyse the convergence of this integral:
$$\int_{1}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$ where $p\in \mathbb{R}$.
I have thought to write:
$$\int_{1}^{c}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}+\int_{c}^{+\infty}\frac{\ln(1+x^p)}{\sqrt{x^2-1}}$$
For the second integral, I know that $\frac{\ln(1+x^p)}{\sqrt{x^2-1}}<\frac{(1+x^p)}{\sqrt{x^2-1}}$ so I can study the second one:
if p>0:
$1+x^p\sim x^p$, $\sqrt{x^2-1}\sim x$, so I obtain: $\frac{(1+x^p)}{\sqrt{x^2-1}}\sim \frac{1}{x^{1-p}}$ that converges if p<0. But I have said that p must be >0, so I don't obtain solutions.
If p<0 the intergral function that I use to compare it with the mine, diverges. So, for the theorem on asymptotic comparision, my integral diverges.
But I don't know how to bring out something about the first integral....