Determine the values of $p$ and $q$ for which a certain integral converges

Question

How can I find for which $p$ and $q$ following integral converges: $$\int_1^{\infty}\frac{dx}{x^p+x^q}$$

I am not sure how to arrange denominator.

Answer 1

If $p>1$ then you have

$$\int_1^{\infty} \frac{1}{x^p+x^q} \; dx \leq \int_1^{\infty} \frac{1}{x^p} \; dx$$

which converges. Similarly, if $q>1$.

If neither $p$ nor $q$ is bigger than one, you have

$$\int_1^{\infty} \frac{1}{x^p+x^q} \; dx \geq \int_1^{\infty} \frac{1}{x+x} \; dx$$

which diverges.