I am to find for which values of p and q the following integral converges: $$\int_0^\infty \frac{x^p}{1+x^q}\,dx\quad (q>0)$$
As I tested the limit of the above function with $\frac{x^p}{x^q}$, and found it was $1$, I let myself separate the boundaries from $0$ to $1$, and from $1$ to infinity of the later function.
From $0$ to $1$ it's a normal integral, that leaves us to check what the divergence of the integral from $1$ to infinity, which by the comparison test happens as $q>$$1$+$p$
Thanks for the quick replay.
Hint : Use the fact that $\zeta(k)=\sum_{n=1}^\infty\frac1{n^k}$ diverges for $k\leqslant1\iff k=q-p>1$ .