Why is $\int_{0}^{\infty} \frac{1}{1 + x^t} \mathop{\mathrm{d}x}=\frac{\pi/t}{\sin(\pi/t)}$?
Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only