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Can anyone figure out how to prove that the $$\int_{\sqrt x}^x \frac{1}{\log^2(t)} dt = O\left(\frac{x}{\log^2(x)}\right)$$

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  • $\begingroup$ Your RHS cannot depend on t $\endgroup$ – phaedo Nov 2 '18 at 0:53
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HINT: use $ (1/\log x)' = -1/(x\log^2 x) $ to "solve" the integral, then have a look at LogIntegral

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  • $\begingroup$ well the next part of the question uses Li(x) and asks to prove it so I cant exactly use it to prove the above question. is there another way? $\endgroup$ – sam Nov 2 '18 at 7:11
  • $\begingroup$ you could try to prove that the ratio converges to a nonzero constant, perhaps with the help of en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule $\endgroup$ – phaedo Nov 2 '18 at 12:50

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