$\lim\limits_{n \rightarrow \infty}\left[\frac{1}{1+n^3}+\frac{4}{8+n^3}+\frac{9}{27+n^3}+...+\frac{1}{2 n}\right]=$

$\frac{1}{3} \log 2$

$\lim\limits_{n \rightarrow \infty}\left[\frac{1}{1+n^3}+\frac{4}{2^3+n^3}+.......+\frac{n^2}{n^3+n^3}\right]=\lim\limits_{n \rightarrow \infty}\left[\frac{r^2}{r^3+n^3}\right]=\lim\limits_{n \rightarrow \infty} \frac{1}{n}\left[\frac{\frac{r^2}{n^2}}{\left(1+\frac{r^3}{n^3}\right)}\right]$

$=\int\limits_0^1 \frac{x^2}{1+x^3} dx \quad$  let  $1+x^3=t$

$3 x^2 dx=dt \Rightarrow x^2 dx=\frac{d t}{3}$

$\frac{1}{3} \int\limits_1^2 \frac{d t}{t}=\frac{1}{3}|\log t|_1^2=\frac{1}{3} \log 2$

Hence (1) is the correct answer.