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Determine whether the following improper integral is convergent or divergent.

$$\int_1^{\infty} \text{sech}\, x \ln x \,dx$$

I think that I need to use integration by parts but the sechx is really stumping me.

Thanks!

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If $x\ge 1$ then $$0\le\frac{\ln x}{\cosh x}=\frac{2\ln x}{e^x+e^{-x}}\le \frac{2\ln x}{e^x+0}=\frac{2\ln x}{e^x}\le 2xe^{-x}$$ and apply comparison test.

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  • $\begingroup$ I don't understand how lnx/e^x is greater than lnx/coshx. Could you please clarify? $\endgroup$ – Melanie Sep 15 '14 at 2:59
  • $\begingroup$ @Melanie Oh, it is a my mistake. I modify it. $\endgroup$ – Hanul Jeon Sep 15 '14 at 3:00
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Hint:

$$\text{sech}\,x\ln x = \frac{2\ln x}{e^x+e^{-x}}.$$

$\ln x$ can be bounded by $\sqrt{x}$ and $\dfrac{1}{e^x+e^{-x}} \le \dfrac{1}{e^x}.$ Do you see how these help?

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Starting from Tetori'answer, you could even show that $$\int \frac{2\ln x}{e^x}dx=2 \left(\text{Ei}(-x)-e^{-x} \log (x)\right)$$ and so $$\int_1^\infty \frac{2\ln x}{e^x}dx=-2 \text{Ei}(-1)\simeq 0.438768$$ while, numerically,$$\int_1^{\infty} \text{sech}\, x \ln x \,dx \simeq 0.430498$$

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