# Finding all values $p$ for which $\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$ converges

I've been stuck for a while with this exercise. Find all positive real values $$p$$ for which the integral $$\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx$$ converges. So far I've came up with this: $$\int_e^{+\infty} \frac{\ln(x)}{(1+x^3)^\frac{1}{p}}dx\ge\int_e^{+\infty} \frac{1}{(1+x^3)^\frac{1}{p}}dx \\ \text{Take the limit of dividing the second function by }\frac{1}{x^\frac{3}{p}} \\ \lim_{x\to{+\infty}}\frac{x^\frac{3}{p}}{(1+x^3)^\frac{1}{p}}=1\ \\ \int_e^{+\infty}\frac{1}{x^\frac{3}{p}}\text{ Diverges } \leftrightarrow p\ge3$$ So, when $$p\ge3$$ my p-series diverges, which means my lower boundary for the main function diverges, which implies that the main function diverges. But I'm unable to prove any other implication. Any suggestions?

## 1 Answer

Hint. One may use that (with an integration by parts) $$\int_e^\infty \frac{\ln x}{x^a}\,dx \qquad \text{converges iff} \qquad a>1.$$ then one may observe that, as $$x \to \infty$$, $$\frac{\ln x}{(1+x^3)^\frac{1}{p}} \sim \frac{\ln x}{x^{3/p}}.$$