Evaluating the double integral $\int_0^\infty d a \int_0^\infty d b\ \frac{ \sin(x a) \sin( y b ) }{a+b}$.

$$I=\int_{0}^{\infty} \int_{0}^{\infty} \frac{\sin ax \sin bx}{a+b} da db = \int_{0}^{\infty} \int_{0}^{\infty} \sin ax \sin by ~ da ~ db ~ e^{-(a+b)t}~dt$$ $$I=\int_{0}^{\infty} dt\left(\int_{0}^{\infty} \sin ax ~e^{-ta} da \int_{0}^{\infty}\sin by ~ db ~ e^{-tb}\right)$$ $$I=\int_{0}^{\infty} \frac{xy}{(t^2+x^2)(t^2+y^2)}dt= \frac{xy}{y^2-x^2} \int_{0}^{\infty}\left(\frac{1}{t^2+x^2}-\frac{1}{t^2+y^2}\right)=\frac{\pi}{2(x+y)}.$$ Note that: $$J=\int \sin ax ~e^{-ta} da=\Im \int_{0}^{\infty} e^{-(t-ix)}=\frac{x}{t^2+x^2}$$