For what $p \in \mathbb{R}$ does this integral converge?

Question

For what $p \in \mathbb{R}$ is $$\int_0^{+\infty} \frac{1}{1+x^p}dx$$ convergent?

I'm confused because I know that $\int_0^{1} \frac{1}{x^p}dx$ is convergent for $p < 1$ and $\int_1^{+ \infty} \frac{1}{x^p}dx$ for $p > 1$. Does this mean that the above integral never converges?

Answer 1

$\int_0^{1}\frac 1{1+x^{p}} dx\leq \int_0^{1} 1dx <\infty$ for all $p>0$. Now use your result for convergence of $\int_1^{\infty}\frac 1{1+x^{p}} dx$.