Question: Let $f$ be a measurable function on $[0,1]$ such that $$g(x) = |f(x)| \ln(1+x)$$ is Lebsegue integrable over $[0,1].$ Prove that $f$ is Lebesgue integrable over $[0,1]$
So the question is asking us to show that $$\int_0^1 |f(x)|\, dx = \int_0^1 \frac{|g(x)|}{\ln(1+x)}dx$$ is finite.
However, I think that the question is not correct. Clearly $g(x) = 1$ is Lebesgue integrable over $[0,1],$ as $$\int_0^1 |g(x)| \, dx = 1,$$ but $$\int_0^1 \frac{1}{\ln(1+x)}dx,$$ based on Wolfram Alpha, is not finite. So $f$ is not Lebesgue integrable over $[0,1].$
Is my reasoning correct?
Your reasoning is correct: intuitively, it could be thanks to the fact that $\ln(1+x)$ goes to $0$ as $x$ goes to zero that $g$ is integrable while $f$ alone is not.
The example $f\colon x\mapsto 1/\ln(1+x)$ shows this. It is indeed true that $\int_0^1 \frac{1}{\ln(1+x)}\mathrm dx$ is divergent but this requires a mathematical justification. This is due to the fact that $\lim_{x\to 0}\ln(1+x)/x=1$ and the divergence of $\int_0^1 \frac{1}{x}\mathrm dx$.