Does the integral of ln(x)/(1+exp(x)) from 2 to ∞ converge or diverge?

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The integral of ln(x)/(1+exp(x)) from 2 to ∞ is being analyzed for convergence or divergence. The discussion clarifies that ln(x) is the intended function, and a comparison test is suggested for evaluation. It is noted that ln(x)/(1+exp(x)) is less than x/(1+exp(x)), which in turn is less than x/exp(x). The integral of x/exp(x) is established to converge, indicating that the original integral also converges. Thus, the integral from 2 to ∞ converges.
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determine whether the integral In(x)/(1+exp(x)) from 2 to ∞? converges or diverges?
 
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What is this function In(x)? Do you mean ln(x)? Assuming so, I think you can do a simple comparison test:

0 < ln(x) / (1+exp(x)) < x / (1+exp(x)) < x / exp(x)

And it's fairly easy to show the integral of x/exp(x) converges.
 

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