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I've been trying to figure out how to solve this: $$\int\limits_{2}^{\infty} \frac{x}{(x+1)\ln(x)}dx = \int\limits_{2}^{\infty} f(x) dx$$ My approach has been to define a $g(x) \ : \ 0 \leq f(x) \leq k g(x)$ as $g(x)=\frac{1}{ln(x)}$ since $\frac{x}{x+1}$ goes to $1$ as $x \rightarrow \infty$.

I've found that $\int\limits_{2}^{\infty} \frac{1}{ln(x)}=li(x)$ that diverges, BUT, are there different ways to solve this? Like ways to get that the integral of ln(x) diverges, or different approaches that doesn't need to answer the former question?

Тyma Gaidash
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matteo.fazio
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    No, $\int_2^{\infty}\frac1{\ln x},dx$ does not exist, and if it did, it would be a constant, not a function of $x.$ – Thomas Andrews Oct 16 '21 at 14:46
  • Write your $x$ in the numerator as $(x+1)-1$? – Cameron Williams Oct 16 '21 at 14:46
  • $\frac{x}{(x+1)ln(x)}=\frac{1}{ln(x)}-\frac{1}{(x+1)ln(x)}$ ? – user408858 Oct 16 '21 at 14:47
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    $x/(x+1)$ doesn't go to $0$. It's greater than $1/2$ for all $x$ in your interval. So your integrand is greater than $1/(2 \ln x)$ and so the integral diverges. – B. Goddard Oct 16 '21 at 14:48
  • In any event, your function $f(x)>\frac1{2x},$ and what is $\in_2^\infty \frac1{2x},dx?$ – Thomas Andrews Oct 16 '21 at 14:48
  • @B.Goddard oh yeah sorry, it was a typo thx, btw your conclusion as the others are what I'm trying to avoid since we use the "oh yeah the integral of 1/ln(x) goes to infinity" without proving it – matteo.fazio Oct 16 '21 at 15:09
  • @ThomasAndrews yeah you're right, I meant that the indefinite integral* is the function li(x), that diverges as $x \rightarrow \infty$ – matteo.fazio Oct 16 '21 at 15:11
  • @ThomasAndrews the premise is correct, $f(x)>\frac{1}{2x}$. Said this you can't conclude that it converges just because a smaller function converges, it'd apply only if you find a greater* function that converges – matteo.fazio Oct 16 '21 at 15:12
  • @matteo.fazio who said the smaller one converges? Glad you could read the comment with my typo of $\in$ instead of $\int.$ – Thomas Andrews Oct 16 '21 at 15:26
  • @ThomasAndrews ohhh wait, you're so right ahah, it diverges, thank you. – matteo.fazio Oct 16 '21 at 15:49

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