(i) Evaluate

\[\int_{[0, \infty)^{2}} \frac{d x d y}{(1+y)\left(1+x^{2} yight)}\]

(ii) Use (i), in order to evaluate

\[\int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x\]

(iii) Use a series representation in part (ii) to show that

\[\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8}\]