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If you take the counter-clockwise contour of a closed semi-circle with radius $R$ in the complex plane with a contour along the real line from $-R$ to $+R$, we have
$$I_0=\int_{-\infty}^\infty\frac{1-x}{1-x^5}\ dx=P.V.\int_{-\infty}^\infty\frac{1-x}{1-x^5}\ dx$$
$$\lim_{R\to\infty}\oint_{\gamma...