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Question Number 161919 by talminator2856791 last updated on 24/Dec/21

∫_(−∞) ^( ∞) sin(x^2 +x+1)dx

$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{−\infty} ^{\:\infty} \:\mathrm{sin}\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right){dx}\: \\ $$$$\: \\ $$

Answered by Lordose last updated on 24/Dec/21

Ω = ∫_(−∞) ^( ∞) sin(x^2 +x+1)dx Ω = ∫_(−∞) ^( ∞) sin((x+(1/2))^2 +(3/4))dx Ω = cos((3/4))∫_(−∞) ^( ∞) sin((x+(1/2))^2 )dx + sin((3/4))∫_(−∞) ^( ∞) cos((x+(1/2))^2 )dx Ω =^(x=x+(1/2)) cos((3/4))∫_(−∞) ^( ∞) sin(x^2 )dx + sin((3/4))∫_(−∞) ^( ∞) cos(x^2 )dx Ω = 2cos((3/4))∫_0 ^( ∞) sin(x^2 )dx + 2sin((3/4))∫_0 ^( ∞) cos(x^2 )dx ∫_0 ^( ∞) sin(x^2 )dx = ∫_0 ^( ∞) cos(x^2 )dx = (√(𝛑/8)) Ω = 2cos((3/4))∙(√(𝛑/8)) + 2sin((3/4))(√(𝛑/8)) Ω = (√(𝛑/2))(cos((3/4))+sin((3/4)))

$$ \\ $$$$\Omega\:=\:\int_{−\infty} ^{\:\infty} \mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)\mathrm{dx} \\ $$$$\Omega\:=\:\int_{−\infty} ^{\:\infty} \mathrm{sin}\left(\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}\right)\mathrm{dx} \\ $$$$\Omega\:=\:\mathrm{cos}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{−\infty} ^{\:\infty} \mathrm{sin}\left(\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \right)\mathrm{dx}\:+\:\mathrm{sin}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{−\infty} ^{\:\infty} \mathrm{cos}\left(\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \right)\mathrm{dx} \\ $$$$\Omega\:\overset{\mathrm{x}=\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}} {=}\mathrm{cos}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{−\infty} ^{\:\infty} \mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\:+\:\mathrm{sin}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{−\infty} ^{\:\infty} \mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$$$\Omega\:=\:\mathrm{2cos}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{\mathrm{0}} ^{\:\infty} \mathrm{sin}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}\:+\:\mathrm{2sin}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\int_{\mathrm{0}} ^{\:\infty} \mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \boldsymbol{\mathrm{sin}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{dx}}\:=\:\int_{\mathrm{0}} ^{\:\infty} \boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\boldsymbol{\mathrm{dx}}\:=\:\sqrt{\frac{\boldsymbol{\pi}}{\mathrm{8}}} \\ $$$$\Omega\:=\:\mathrm{2cos}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\centerdot\sqrt{\frac{\boldsymbol{\pi}}{\mathrm{8}}}\:+\:\mathrm{2sin}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\sqrt{\frac{\boldsymbol{\pi}}{\mathrm{8}}} \\ $$$$\Omega\:=\:\sqrt{\frac{\boldsymbol{\pi}}{\mathrm{2}}}\left(\mathrm{cos}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)+\mathrm{sin}\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\right) \\ $$$$ \\ $$

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