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Question Number 128110 by Dwaipayan Shikari last updated on 04/Jan/21

$$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{sinx}\:{sin}\left({x}+{y}\right)}{{x}\left({x}+{y}\right)}{dxdy} \\$$

Answered by Olaf last updated on 04/Jan/21

$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}\mathrm{sin}\left({x}+{y}\right)}{{x}\left({x}+{y}\right)}{dxdy} \\$$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}\left(\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}\left({x}+{y}\right)}{{x}+{y}}{dy}\right){dx} \\$$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}\left(\int_{{x}} ^{\infty} \frac{\mathrm{sin}{u}}{{u}}{du}\right){dx} \\$$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}\left(\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{u}}{{u}}{du}−\int_{\mathrm{0}} ^{{x}} \frac{\mathrm{sin}{u}}{{u}}{du}\right){dx} \\$$$$\Omega\:=\:\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}\left(\frac{\pi}{\mathrm{2}}−\mathrm{Si}\left({x}\right)\right){dx} \\$$$$\Omega\:=\:\frac{\pi}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}{dx}−\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sin}{x}}{{x}}\mathrm{Si}\left({x}\right){dx} \\$$$$\Omega\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−\int_{\mathrm{0}} ^{\infty} \mathrm{Si}'\left({x}\right)\mathrm{Si}\left({x}\right){dx} \\$$$$\Omega\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{Si}^{\mathrm{2}} \left({x}\right)\right]_{\mathrm{0}} ^{\infty} \\$$$$\Omega\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{Si}\left({x}\right)\right)^{\mathrm{2}} \\$$$$\Omega\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{4}}−\frac{\mathrm{1}}{\mathrm{2}}.\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\$$$$\Omega\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\$$

Commented by Dwaipayan Shikari last updated on 04/Jan/21

$${Great}\:{sir}\:! \\$$

Commented by BHOOPENDRA last updated on 05/Jan/21

$${great}\:{sir} \\$$