I'm trying to understand the application of residue calculus to a worked example given in subsection 7.8 of this online document. The example gives two methods of evaluating the integral $I=\int_0^\infty\frac{dx}{1+x^3}.$ I'm looking at method 1 only (subsection 7.8.1) in which the author asks the reader to "...consider the related integral $\oint_C\frac{\log z}{1+z^3}dz,$ over the contour shown in Fig. 7.6." (see the PDF for figure). However, I do not grasp why in the contour integral we would choose the given integrand since it is different to that in the first integral above.
Any hints or an answer would be helpful to see what's going on. Many thanks.