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Question Number 215061 by mnjuly1970 last updated on 27/Dec/24

$$$$$${\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{x^2+y^2}{2}}\sin \left(xy\right)dxdy=?$$$$$$

Answered by MrGaster last updated on 27/Dec/24

$$I={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x^2+y^2}{2}}\sin \left(xy\right)dxdy$$$$={\int }_0^{\frac{\pi }{2}}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{r^2}{2}}\sin \left(\frac{r^2}{2}\sin 2\theta \right)rdrd\theta$$$$={\int }_0^{\frac{\pi }{2}}\frac{\sin 2\theta }{1+{\sin }^{2}2\theta }d\theta$$$$=\frac{1}{4\sqrt{2}}\ln (17+12\sqrt{2})$$

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