Let $R$ be a rational function such that $\lim _{z \rightarrow \infty}\{z R(z)\}=0$. Assuming that $R$ has no real poles, use the residue calculus to evaluate

$$\int_{-\infty}^{\infty} R(x) d x$$

Given that $n \geqslant 1$ is an integer, evaluate

$$\int_{0}^{\infty} \frac{d x}{1+x^{2 n}}$$