Determine if $\int_1^{\infty}\frac{dx}{x^p+x^q}$ converges ...

Question

Determine if $\int_1^{\infty}\frac{dx}{x^p+x^q}$ converges if $\min(p, q) < 1$ and $\max(p, q) > 1$, where $\min (p, q)$ is the minor of the numbers $p$ and $q$, and $\max (p,q)$ is the major of the numbers $p$ and $q$.

I have doubts of how to arrange denominator.

Answer 1

Suppose $p<q$. Then $$\frac{1}{x^p+x^q}= \frac{1}{x^q(1+x^{p-q})} \sim \frac{1}{x^q} $$ So integral converges when $q>1$.

It is interesting consider this integral on whole $(0, +\infty)$, so for convergence should add conditions from existence $$\int_{0}^{1}\frac{dx}{x^p+x^q} $$ Obviously integral for $(0, +\infty)$ diverges when $p=q$, so make sense consider $p \ne q$. For second case we have $$\frac{1}{x^p+x^q}= \frac{1}{x^p(1+x^{q-p})}$$ and obtain, that integral diverges for $p<1$.

So conditions for whole $(0, +\infty)$, when $p<q$, are $p<1$ and $q>1$.

Answer 2

Assume that $p\ge q$.

then , when $ x\to +\infty,$

$$x^p+x^q=x^p(1+x^{q-p})$$ $$\sim x^p \; \text{ or } \; 2x^p$$

the integral converges if $ p>1 $.

we conclude that the integral converges if $ \max(p,q)>1$, diverges if $\max(p,q)\le 1$.