Question Number 215061 by mnjuly1970 last updated on 27/Dec/24
$$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \mathrm{e}^{\:\:−\:\frac{\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{sin}\left({xy}\:\right){dxdy}=? \\ $$$$ \\ $$
Answered by MrGaster last updated on 27/Dec/24
$${I}=\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} {e}^{−\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{sin}\left({xy}\right){dxdy} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\infty} {e}^{−\frac{{r}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{sin}\left(\frac{{r}^{\mathrm{2}} }{\mathrm{2}}\mathrm{sin2}\theta\right){rdrd}\theta \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{1}+\mathrm{sin}^{\mathrm{2}} \mathrm{2}\theta}{d}\theta \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}\sqrt{\mathrm{2}}}\mathrm{ln}\left(\mathrm{17}+\mathrm{12}\sqrt{\mathrm{2}}\right) \\ $$
Commented by MathematicalUser2357 last updated on 27/Dec/24
$$\mathrm{0}.\mathrm{62322}\:\mathrm{52401}\:\mathrm{40230}\:\mathrm{51339}\:\mathrm{42001} \\ $$