By the definition of the definite integral, the value of $\lim _{n \rightarrow \infty}\left[\frac{1^2}{1^3+n^3}+\frac{2^2}{2^3+n^3}+\ldots+\frac{r^2}{r^3+n^3}+\ldots+\frac{1}{2 n}\right]=$

• A

  $\sqrt[4]{\log 2}$
• Correct answer
  B

  $\log \sqrt[3]{2}$
• C

  $\sqrt{\log 2}$
• D

  $\log \sqrt[3]{3}$

Hence, the correct answer is option: 'B'.