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Another real integral using contour integration.

[Click here for a PDF version of this post] Here’s (31(d)) from [1]. Find \begin{equation}\label{eqn:fourPoles:20} I = \int_0^\infty \frac{dx}{1 + x^4} = \inv{2}\int_{-\infty}^\infty \frac{dx}{1 + x^4}. \end{equation} This one is easy conceptually, but a bit messy algebraically. We integrate over the contour \( C \) illustrated in fig. 1. fig. 1. Standard above the x-axis, semicircular…