November 21st, 2018, 08:06 AM  #1 
Newbie Joined: Oct 2015 From: algeria Posts: 5 Thanks: 0  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. Thank you 
November 21st, 2018, 09:54 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 
It certainly looks like the integral blows up at multiples of $\pi$, i.e. where $\sin^2(x)=0$ 
November 21st, 2018, 01:41 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,807 Thanks: 716 
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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function, improper, integral 
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