integration - How to evaluate $lim_{xto +infty} int_x^{x^3} frac{dt}{(ln(t))^2}$?

I'm searching $\lim_{x\to +\infty} \int_x^{x^3} \frac{dt}{(ln(t))^2}$, but I'm stuck.

I've tried to do a change of variable in order to get $u\to 0$ and then use a Taylor expansion... But nothing works.

Answer

Since $\log t$ is an increasing function, we have:

$$I(x) = \int_{x}^{x^3}\frac{dt}{\log^2 t} \geq \frac{x^3-x}{\log^2(x^3)}$$
so, simply,
$$\lim_{x\to +\infty} I(x) = +\infty.$$