My goal is to evaluate $$\int_{0}^{\infty}\frac{dx}{1+x^n}\;\;(n\in\mathbb{N},n\geq2).$$
Here is my approach:
Clearly, the integral converges.
Denote the value of the integral by $I_n$.
Now let $\gamma_R$ describe the section of a circle which goes from the origin to $R$ to $Re^{2\pi i/n}$ and back to the origin.
If we let $C_R$ denote the relevant circular arc, then $$\left|\int_{C_R}\frac{dz}{1+z^n}\right|\leq \left(\frac{2\pi R}{n}\right)\left(\frac{1}{R^{n}-1}\right)\rightarrow0\;\;\;as\;\;R\rightarrow\infty.$$
Furthermore, $$\int_{[R,Re^{2\pi i/n}]}\frac{dz}{1+z^n}=\int_{R}^{0}\frac{e^{2\pi i/n}dr}{1+r^n}.$$
Hence $$\lim_{R\rightarrow\infty}\int_{\gamma_R}\frac{dz}{1+z^n}=\lim_{R\rightarrow\infty}\int_{[0,R]}\frac{dx}{1+x^n}+\int_{[R,Re^{2\pi i/n}]}\frac{dx}{1+x^n}+\int_{C_R}\frac{dx}{1+x^n}=(1-e^{2\pi i/n})I_n\;\;\;(1).$$
Thus if we can obtain the value of $\int_{\gamma_R}\frac{dz}{1+z^n}$ we can evaluate $I_n$.
Now the zeroes of $1+z^n$ are of the form $z=e^{i\pi/n+2\pi i m/n}\;\;(m\in\mathbb{N})$ from which it is clear that the only zero which lies within the contour occurs at $z=e^{i\pi/n}$ with multiplicity 1. So all that remains to be done is to evaluate the residue of $\frac{1}{1+z^n}$ at $z=e^{i\pi/n}$.
However, if $z=e^{i\pi/n}u$ and $u\neq1$, we have $$\frac{z^n+1}{z-e^{i\pi/n}}=\frac{1-u^n}{-e^{i\pi/n}(1-u)} =-e^{-i\pi/n}\sum_{m=0}^{n-1}u^m\;\;\;(2).$$
In particular, (2) implies $$Res_{z=e^{i\pi/n}}\frac{1}{1+z^n}=-\frac{e^{i\pi/n}}{n}\;\;\;(3).$$
Finally, (1) and (3) imply $$I_n=\frac{2\pi i (Res_{z=e^{i\pi/n}}\frac{1}{1+z^n})}{1-e^{2\pi i/n}}=\frac{-2\pi ie^{i\pi/n}}{n(1-e^{2\pi i/n})}=\frac{\pi/n}{\sin(\pi/n)}.$$
I have three questions:
One, is my method correct?
Two, is there a simpler/different method to evaluate the integral?
Three, is there an easier way to evaluate the residue of $\frac{1}{1+z^4}$ at $z=e^{i\pi/n}$?
