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Question Number 689 by 112358 last updated on 25/Feb/15

Evaluate   (1)               ∫_0 ^∞ sin(x^2 )dx,  (2)               ∫_0 ^∞ cos(x^2 )dx,  (3)               ∫_0 ^∞ tan(x^2 )dx.

$${Evaluate}\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{\mathrm{2}} \right){dx}, \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {cos}\left({x}^{\mathrm{2}} \right){dx}, \\ $$$$\left(\mathrm{3}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} {tan}\left({x}^{\mathrm{2}} \right){dx}. \\ $$

Commented by 123456 last updated on 25/Feb/15

 ∫_Γ e^(ız^2 ) dz

$$\:\underset{\Gamma} {\int}{e}^{\imath{z}^{\mathrm{2}} } {dz} \\ $$

Answered by prakash jain last updated on 25/Feb/15

We know that  ∫_0 ^∞ e^(−x^2 ) dx=((√π)/2)  sin (x^2 )=−Im (e^(−ix^2 ) )  cos (x^2 )=Re (e^(−ix^2 ) )  ∫_0 ^∞ e^(−ix^2 ) dx=(1/(√i))∙((√π)/2)=((1/(√2)) −(i/(√2)))∙((√π)/2)  ∫_0 ^∞ sin (x^2 )=((√π)/(2(√2)))  ∫_0 ^∞ cos (x^2 )=((√π)/(2(√2)))

$$\mathrm{We}\:\mathrm{know}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$$$\mathrm{sin}\:\left({x}^{\mathrm{2}} \right)=−\mathrm{Im}\:\left({e}^{−{ix}^{\mathrm{2}} } \right) \\ $$$$\mathrm{cos}\:\left({x}^{\mathrm{2}} \right)=\mathrm{Re}\:\left({e}^{−{ix}^{\mathrm{2}} } \right) \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{ix}^{\mathrm{2}} } {dx}=\frac{\mathrm{1}}{\sqrt{{i}}}\centerdot\frac{\sqrt{\pi}}{\mathrm{2}}=\left(\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}\:−\frac{{i}}{\sqrt{\mathrm{2}}}\right)\centerdot\frac{\sqrt{\pi}}{\mathrm{2}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \mathrm{sin}\:\left({x}^{\mathrm{2}} \right)=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$$$\int_{\mathrm{0}} ^{\infty} \mathrm{cos}\:\left({x}^{\mathrm{2}} \right)=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{\mathrm{2}}} \\ $$

Commented by prakash jain last updated on 25/Feb/15

∫_0 ^∞ tan (x^2 ) dx does not coverge.

$$\int_{\mathrm{0}} ^{\infty} \mathrm{tan}\:\left({x}^{\mathrm{2}} \right)\:{dx}\:\mathrm{does}\:\mathrm{not}\:\mathrm{coverge}.\: \\ $$

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