I have the integral $$\int_2^\infty\frac{3}{\sqrt[3]x(x+2\sqrt x)}dx$$ and have to find out whether it's divergent or convergent using the comparison test. I've been trying to understand this topic but when I have more than 2 x's in the denominator like here, I have no idea what to compare it to. So far I have $$\int_2^\infty\frac{1}{x^{4/3}+2x^{5/6}}dx$$ so, I tried to compare it to both $\frac{1}{x^{4/3}}$ and $\frac{1}{x^{5/6}}$ but in both cases p<2, which would mean that the integral is divergent. Yet the integral calculator says that there is a result so it should be convergent. I don't know what I'm doing wrong and if anyone could explain it to me, I would be thankful.
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"which would mean" why? $$\int_{2}^{+\infty}\frac{dx}{x^p}$$ is convergent as soon as $p>\color{red}{\large 1}$. In particular,
$$\int_{2}^{+\infty}\frac{dx}{x^{4/3}}=\frac{3}{\sqrt[3]{2}},$$ hence your integral is convergent.
answered Jun 22, 2016 at 17:18