What is $\int_b^\infty\frac{\ln x}{x^2+1}dx$? 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.
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
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