$$\int_0^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx\quad\quad a > b > 0$$
I have no idea how to compute this. Any help would be greatly appreciated.
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Don't know if helps but I would start with partial fractions: $$\int\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx=\int\frac{x \sin x }{(a^2-b^2) (b^2+x^2)}dx+\int\frac{x \sin x }{(b^2-a^2) (a^2+x^2)}dx$$ – rlartiga Nov 28 '14 at 02:22
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Hints:
Evaluate $~J(k)=\displaystyle\int_0^\infty\frac{\cos(kx)}{x^2+n^2}dx,~$ and then express the integral(s) in terms of $J'(k)$.
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notice the integrand is an even function, so you may take half of the integral along the entire real axis. since, as a function of a complex variable, it is also $O(|z|^{-2})$ as $z \rightarrow \infty$, the integral round a large semicircle $\rightarrow 0$, so you may use the residue theorem with poles $z=bi$ and $z=ai$
this gives $$ \int_{-\infty}^\infty\frac{x \sin x }{(x^2 + a^2)(x^2 + b^2)}dx = \frac{\sinh b - \sinh a}{(b^2-a^2)}\pi $$
David Holden
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