Q. Determine the values of p and q for which the improper integral converges \begin{equation} \int_0^{\infty}x^p[\ln(1+x)]^q\,dx \end{equation}
After some calculation, I found out this integral diverges when:
$p,q \leq 0$ or $p \geq 0, q \leq 0$ or $p,q \geq 0$
So we have to only consider when $p < 0, q > 0 $ .
I used integration by parts method but after that, the remaining integral has a different form. For example: \begin{equation} \int_2^{\infty}x^p(\ln x)^q\,dx \end{equation}
This case is rather easy because after using integration by parts, the remaining integral also has a same form, that is:
\begin{equation} {q\over p+1}\int_2^{\infty}x^p(\ln x)^{q-1}\,dx \end{equation}
How do I proceed from here?