A nice contour integral

Consider the contour:

\displaystyle C_R = \left\{Re^{i t}, 0 \le t \le \frac {2 \pi} n\right\} \cup \left\{Re^{\frac {2 \pi i} n} t, 0 \le t \le 1\right\} \cup \{R t, 0 \le t \le 1\}

(this is a sector of a circle centred at the origin with angle \displaystyle \frac {2 \pi} n, in the first quadrant, with radius on the real axis)

By integrating around C_R, show that:

\displaystyle \int_0^\infty \frac 1 {1 + x^n} \mathrm dx = \frac \pi n \csc \frac \pi n