(i) A function $f(z)$ has a pole of order $m$ at $z=z_{0}$. Derive a general expression for the residue of $f(z)$ at $z=z_{0}$ involving $f$ and its derivatives.

(ii) Using contour integration along a contour in the upper half-plane, determine the value of the integral

$$I=\int_{0}^{\infty} \frac{(\ln x)^{2}}{\left(1+x^{2}\right)^{2}} \mathrm{~d} x$$