A) \[\frac{1}{2}\log 2\]
B) \[x=\frac{3\pi }{4}\]
C) \[\pi /4\]
D) \[\pi /2\]
Correct Answer: A
Solution :
Let \[I=\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\,\frac{k}{{{n}^{2}}+{{k}^{2}}}}\] \[=\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{k=1}^{n}{{}}\frac{1}{n}\frac{\left( \frac{k}{n} \right)}{1+{{\left( \frac{k}{n} \right)}^{2}}}\] \[I=\int\limits_{0}^{1}{\frac{x}{1+{{x}^{2}}}dx}\] \[=\frac{1}{2}[\log (1+{{x}^{2}})]_{\,0}^{\,1}\]\[=\frac{1}{2}\left[ \log 2 \right]\].
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