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How can I evaluate $$\int_0^{\infty}\frac{x\sin(bx)}{x^2+1}dx \quad(b>0)$$ using residue calculus? I tried computing the residue of $\int_0^{\infty}\frac{z\exp(ibz)}{z^2+1}dz$ at $z=i$ and taking the imaginary part of it, but what contour should I use so that only the integral over the real axis is left?

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  • $\begingroup$ What is $b$? If it is a positive real number then take half a circle. The complex function is $\frac{ze^{ibz}}{z^2+1}$. To bound it on the circle path you will probably need Jordan's lemma. $\endgroup$ – Mark Jun 13 at 19:36
  • $\begingroup$ @Mark How can it be a semicircle if the limit is from 0 to infinity? Won't it be a quarter of a circle? $\endgroup$ – ranger281 Jun 13 at 19:43
  • $\begingroup$ Yes, but the real function you integrate is even, so you can just find the integral from $-\infty$ to $\infty$ and then divide by $2$ to get the integral from $0$ to $\infty$. $\endgroup$ – Mark Jun 13 at 19:44
  • $\begingroup$ Why is it even if there's x in the numerator? $\endgroup$ – ranger281 Jun 13 at 19:48
  • $\begingroup$ Did you forget the $\color{blue}{\sin}$? $\endgroup$ – metamorphy Jun 13 at 19:59

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