Question Number 128567 by mnjuly1970 last updated on 08/Jan/21
$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:{prove}\:{that}:::: \\ $$$$\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} {sin}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{−\mathrm{2}} \right){dx} \\ $$$$\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\frac{\pi}{\mathrm{2}}}\:\left({e}^{−\mathrm{2}} +{sin}\left(\mathrm{2}\right)−{cos}\left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$