I'd like your help with checking whether $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3+x}}$ converges or not. Here are the steps which led me to conclude that the integral does converge, but I'm not really sure.
First, I saw that I can't easily compute $\int \frac{dx}{\sqrt{x^3+x}}$ so I wanted to see what happens when x tends to infinity. It's not really formal but I can see that the integral "acts" as $\int_{0}^{\infty} \frac{dx}{\sqrt{x^3}}$ so in the infinity it is almost as $\int_{0}^{\infty} \frac{dx}{x^{3/2}}$ and since it's in the format $\int_{0}^{\infty}\frac{1}{x^p}$ where $p>1$ I concluded that it converges. (Is it true? or do I must separate it to two integrals $\int_{0}^{a}\frac{1}{x^p}$ + $\int_{a}^{\infty}\frac{1}{x^p}$ and check both?).
What do you think? what is the correct way to check convergence for this integral?
Thanks a lot!