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November 21st, 2018, 09:06 AM   #1
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Improper integral of a function

I want to study the nature of an improper integral. I separate the integral from 0 to 1 and from 1 to ∞. The first integral is divergent (I proved that it is equivalent to Riemann integral, which is divergent in the neighborhood of 0). But I didn't find how I can prove the convergence or divergence of the second one. The integral can be found in attach file.
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November 21st, 2018, 10:54 AM   #2
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It certainly looks like the integral blows up at multiples of $\pi$, i.e. where

$\sin^2(x)=0$
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November 21st, 2018, 02:41 PM   #3
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It is not divergent at x=0, since $log(1+x)^2\approx x^2$ there. However as romsek points out it will blow up at multiples of $\pi$.
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