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They ask me to evaluate $$\int_b^\infty\frac{\ln x}{x^2+1}dx, \quad b\geq 0$$

Any suggestions please.

This is in the context of indefinite integrals that do not involve functions such as Li, constant Catalan G, only the elementary functions, even before they teach us Series.

It's important because it's a partial college exam question and I do not find suggestions for any $ b $, just for $ b = 1 $ it's easy to see that the integral mentioned is zero.

I tried to solve it with the method of integration by parts, but it does not work out, that is why I come to your help.

mathsalomon
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  • The improper integral can be evaluated in terms of dilogarithms by expanding the denominator in partial fractions, then doing integration by parts once, then using that $$\text{Li}_2(x)=-\int \frac{\ln(1-x)}x\mathrm dx$$ I'm not sure what to do with the limits – John Doe Apr 25 '19 at 02:01
  • Thanks, but that has not taught us yet. There is no other way to avoid the dilogarithms ? – mathsalomon Apr 25 '19 at 02:40
  • Strongly related: https://math.stackexchange.com/questions/817014/catalans-constant-and-int-1x-frac-logt1t2-dt Especially since $\int_0^1 \frac{\ln x}{x^2+1},\mathrm{d}x = -\int_1^\infty \frac{\ln x}{x^2+1},\mathrm{d}x = -G$, where $G$ is Catalan's constant. – Eric Towers Apr 25 '19 at 02:53
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    @mathsalomon Isn't your integral basically $\int \frac{\arctan x}{x}dx$ (integration by parts), which can be handled via taylor series of $\arctan$? – mathworker21 Apr 25 '19 at 03:06

1 Answers1

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Let $f(x) =\log x$ and $g(x) =\frac{x}{1+x^2}$. Then $f(x) /g(x)$ tends to be $0$ as $x$ tends to infinity. Thus two functions behave alike. But integration $g(x)$ to infinite divergent so $f(x)$ is divergent

Javi
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Ankan Mandal
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