Solve the integral \(\displaystyle\int_0^\infty \frac{1}{1 + x^\alpha}\,dx\) for \(\alpha \in \mathbb{N}\), \(\alpha > 1\) by integrating the function \(\frac{\log z}{1 + z^\alpha}\) along the contour of figure 3.
Figure 3 of the source shows the contour \(C = L_+ \cup C_R \cup L_- \cup \gamma_\epsilon\): \(L_+\) runs just above the positive real axis from the small circle \(\gamma_\epsilon\) around \(0\) out to the large circle \(C_R\) of radius \(R\) centered at \(0\), \(C_R\) is traversed counterclockwise, \(L_-\) returns just below the positive real axis, and \(\gamma_\epsilon\) is traversed clockwise around \(0\). In the source this contour is used with the branch cut on the positive real axis, the radius of \(\gamma_\epsilon\) sent to zero and the radius of \(C_R\) sent to infinity.