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,
$$ \lim_{x\to +\infty} I(x) = +\infty.$$
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