It seems like standard contour integration methods do not work well when integrating from 0 to $\infty$ of odd functions. For example, consider the following integral: $$I=\int_0^\infty \frac{\sin(x)}{x^2+1}dx$$ The usual approach fails since we would like to use a semicircular contour, however the integral along the real line vanishes and we cannot recover the integral $I$. Are there any workarounds using methods of contour integration, perhaps defining a different complex function to integrate? Or any other methods to evaluate this integral?
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You can try to compute the integral of $e^{ix}/(x^2+1)$ first. – richrow Jan 11 '22 at 11:23
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@richrow is right: in this way you compute a Fourier transform and you take its imaginary part at the very end. Different contours are possible for that : see for example here https://math.stackexchange.com/q/4120340 – Jean Marie Jan 11 '22 at 12:29
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@David C. Ullrich : No it deals as well with $\int_{0}^{\infty}$... – Jean Marie Jan 12 '22 at 10:56
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@David C. Ullrich Do we refer to the same question "Tricky contour integral..." ? If yes, see the beginning of the question. – Jean Marie Jan 12 '22 at 11:05