Does $\int_1^\infty f(x)\ln(x)dx$ converge if $\int_1^\infty f(x)dx $ converges?

Question

Suppose a function $f:[1,\infty)\to\mathbb R$ is such that $\int_1^\infty f(x)\,dx $ converges. Is it possible that $$\int_1^\infty f(x)\ln(x)\,dx $$ diverges? I have a hard time finding such a function.
Edit: no idea why, but I had just thought naively (without checking) that $\int \frac 1{x\ln^k(x)}\,dx $ diverges for all $k$ just because $\int \frac 1{x\ln(x)}\,dx $ diverges. Sorry!

Answer 1

Hint. Consider the function $$f(x)=\frac{1}{x\ln^2(1+x)}.$$

Is $\int_1^\infty f(x)dx$ convergent? What about $\int_1^\infty f(x)\ln(x) dx$?