Let

$$f(z)=\frac{z^{m}}{1+z^{n}}, \quad n>m+1, \quad m, n \in \mathbb{N},$$

and let $C_{R}$ be the boundary of the domain

$$D_{R}=\left\{z=r e^{i \theta}: 0<r<R, \quad 0<\theta<\frac{2 \pi}{n}\right\}, \quad R>1 .$$

(a) Using the residue theorem, determine

$$\int_{C_{R}} f(z) d z$$

(b) Show that the integral of $f(z)$ along the circular part $\gamma_{R}$ of $C_{R}$ tends to 0 as $R \rightarrow \infty$.

(c) Deduce that

$$\int_{0}^{\infty} \frac{x^{m}}{1+x^{n}} d x=\frac{\pi}{n \sin \frac{\pi(m+1)}{n}}$$