Prove that $\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n})$

Use contour integrals and residue theory to prove that
$$\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n}).$$ 

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Erdos Erdos
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