Integrate $\frac{dx}{x^{2n}+1}$ for $n \in \mathbb{N}$ from $-\infty$ to $\infty$
Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only