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Question Number 128567 by mnjuly1970 last updated on 08/Jan/21

             ...nice   calculus...   prove that::::     φ=∫_0 ^( ∞) sin(x^2 )sin(x^(−2) )dx       =(1/4)(√(π/2)) (e^(−2) +sin(2)−cos(2))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}... \\ $$$$\:{prove}\:{that}:::: \\ $$$$\:\:\:\phi=\int_{\mathrm{0}} ^{\:\infty} {sin}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{−\mathrm{2}} \right){dx} \\ $$$$\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{4}}\sqrt{\frac{\pi}{\mathrm{2}}}\:\left({e}^{−\mathrm{2}} +{sin}\left(\mathrm{2}\right)−{cos}\left(\mathrm{2}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

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