AP EAMCET · Maths · Definite Integration
\(\lim _{n \rightarrow \infty}\left[\frac{n+1}{n^2+1^2}+\frac{n+2}{n^2+2^2}+\frac{n+3}{n^2+3^2}+\ldots+\frac{n+2 n}{n^2+4 n^2}\right]=\)
- A \(\operatorname{Tan}^{-1} 2+\frac{1}{2} \log 3\)
- B \(\frac{\pi}{4}+\frac{1}{2} \log 3\)
- C \(\operatorname{Tan}^{-1} 2+\frac{1}{2} \log 5\)
- D \(\frac{\pi}{4}+\frac{1}{2} \log 5\)
Answer & Solution
Correct Answer
(C) \(\operatorname{Tan}^{-1} 2+\frac{1}{2} \log 5\)
Step-by-step Solution
Detailed explanation
\( \lim _{n \rightarrow \infty} \sum_{k=1}^{2n} \frac{n+k}{n^2+k^2} = \lim _{n \rightarrow \infty} \sum_{k=1}^{2n} \frac{1}{n} \frac{1+\frac{k}{n}}{1+(\frac{k}{n})^2} \) \( = \int_{0}^{2} \frac{1+x}{1+x^2} dx \)…
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