Evaluate the definite integral (int_{-infty}^{+infty} frac{d x}{left(1+x^{2}ight)}), where the contour runs along the (x) axis from (-ho)

Question:

Evaluate the definite integral \(\int_{-\infty}^{+\infty} \frac{d x}{\left(1+x^{2}ight)}\), where the contour runs along the \(x\) axis from \(-ho\) to \(+ho\), and then closes by a semicircle in the upper half-plane of radius \(ho\) centered at the origin.

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