Complex Methods | Part IB, 2001

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

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

Given that n1n \geqslant 1 is an integer, evaluate

0dx1+x2n\int_{0}^{\infty} \frac{d x}{1+x^{2 n}}

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