I am stuck at the following exercise:
Show that for $n \in \mathbb{N}$ holds
$$\int_2^x \frac{dt}{\log(t)^n} = \mathcal{O}\bigg(\frac{x}{\log(x)^n}\bigg).$$
I do not see how I could prove this. I know that the following identity holds:
$$\int_2^x \frac{dt}{\log(t)^n} = \frac{t}{\log(t)^n} \bigg\vert^x_2 + n \int_2^x\frac{dt}{\log(t)^{n+1}},$$
but I do not see how this could help here. Could you give me a hint?