Problem with finding the limit using integral and summation

Question

Find the limit, interpreting it as a limit of a summation of a suitably chosen function:

$$\lim_{n\to\infty} n \bigg( \frac{1}{n^{2} + 1^{2}} + \frac{1}{n^{2} + 2^{2}} + ... + \frac{1}{n^{2} + n^{2}} \bigg).$$

Answer 1

We have $n*\frac{1}{n^2+k^2}=\frac{1}{n}*\frac{1}{1+(\frac{k}{n})^2}$.

Loook at $f(x)=\frac{1}{1+x^2}$