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\int_0^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx = \frac{\pi}{2}e^{-am}

\]

The inetgral is even so

\[

\frac{1}{2}\int_{-\infty}^{\infty}\frac{x\sin(mx)}{x^2 + a^2}dx.

\]

We can also write \(x^2 + a^2\) as \((x + ai)(x - ai)\). Should I also write \(\sin(mx) = \frac{1}{2i}(z^m - 1/z^m)\)? I tried this but it didn't go anywhere.

\[

\frac{1}{2}\int_{-\infty}^{\infty}\frac{z\sin(mx)}{(z + ai)(z - ai)}dz.

\]

How do I finish this problem?