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Bounding the integral $\int_{2}^{x} \frac{\mathrm dt}{\log^{n}{t}}$
So I want to show that $$\int_2^x \frac{1}{\log^n(t)}\mathrm dt=O\left(\frac{x}{\log^n(x)}\right)$$, and someone suggests that I could split the integral into two parts: $$\int_2^{\sqrt{x}} \frac{1}{\log^n(t)}\mathrm dt+\int_{\sqrt{x}}^x \frac{1}{\log^n(t)}\mathrm dt$$, and why that immediately produces the right error term $O\left(\frac{x}{\log^n(x)}\right)$?