$\lim _{n \rightarrow \infty}\left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\ldots+\frac{n+2 n}{n^2+4 n^2}\right]=$

• A

  $\operatorname{Tan}^{-1} 2+\frac{1}{2} \log 3$
• B

  $\frac{\pi}{4}+\frac{1}{2} \log 3$
• Correct answer
  C

  $\operatorname{Tan}^{-1} 2+\frac{1}{2} \log 5$
• D

  $\frac{\pi}{4}+\frac{1}{2} \log 5$

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