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R-R-I-x-2-ln-1-x-2-x-4-1-e-x-2-cos-x-dx-

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Question Number 220065 by Nicholas666 last updated on 04/May/25
α∈R ; ω∈R^+ I(α) = ∫_(−∞) ^( ∞) ((x^2 ln(1+x^2 ))/((x^4 +1)^α )) e^(−x^2 ) cos(ωx) dx
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\alpha\in\mathbb{R}\:\:\:;\:\:\:\:\omega\in\mathbb{R}^{+} \\ $$$$\:\:\:\:\:{I}\left(\alpha\right)\:=\:\int_{−\infty} ^{\:\infty} \:\frac{{x}^{\mathrm{2}} \:\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\alpha} }\:{e}^{−{x}^{\mathrm{2}} } \:\mathrm{cos}\left(\omega{x}\right)\:{dx} \\ $$$$ \\ $$
Answered by MrGaster last updated on 04/May/25
I have derived three resultsu bt I am not sure which one isc correct (you can verify itr late). (1):⇒I(α)=2∫_0 ^∞ ((x^2 ln(1+x^2 ))/((x^4 +1)^α ))e^(−x^2 ) cos(ωx)dx ln(1+x^2 )=∫_0 ^1 (x^2 /(1+x^2 t))dt ⇒I(α)=2∫_0 ^1 ∫_(0 ) ^∞ (x^4 /((x^4 +1)(1+x^2 t)))e^(−x^2 ) cos(ωx) (x^4 +1)^(−α) =(1/(Γ(α)))∫_0 ^∞ s^(α−1) e^(−(x^4 +1)s) ds ⇒I(α)=(2/(Γ(α)))∫_0 ^1 ∫_(0 ) ^∞ s^(α−1) e^(−s) ∫_0 ^∞ ((x^4 e^(−x^2 (1+sx^2 )) )/(1+x^2 t))cos(ωx)dx ds dt ∫_0 ^∞ x^4 e^(−x^2 (1+sx^2 )) cos(ωx)dx=((√π)/4) (∂^2 /∂λ^2 )((e^(−(ω^2 /(4(1+sx^2 +λ)))) /( (√(1+sx^2 +λ)))))∣_(λ=0) ⇒I(α)=((√π)/(2Γ(α))) (∂^2 /∂λ^2 )(e^(−(ω^2 /(4(1+λ)))) ∫_0 ^∞ ((s^(α−1) e^(−s) )/( (√(1+s+λ))(1+t(1+s+λ))))ds=((Γ(α))/((1+t)^α (√(1+t)))) _2 F_1 (α,(1/2);α+(1/2);(t/(1+t))) I(α)=((√π)/2) (∂^2 /∂λ^2 )((e^(−(ω^2 /(4(1−λ)))) /( (√(1+λ))))∫_0 ^1 (1/((1+t)^α )) _2 F_1 (α,(1/2);α+(1/2);(t/(1+t)))dt)∣_(λ=0) ∫_0 ^1 (( _2 F_1 (α,(1/2);α+(1/2);(t/(1+t))))/((1+t)^α ))dt=(((√π)Γ(α+(1/2)))/(Γ(α+1))) ⇒I(α)=(π/(2Γ(α+1))) (∂^2 /∂λ^2 )((e^(−(ω^2 /(4(1+λ)))) /( (√(1+λ))))Γ(α+(1/2)))∣_(λ=0) ⇒I(α)=((π^(3/2) Γ(α+(1/2)))/(2Γ(α+1)))((ω^4 /4)e^(−(ω^2 /4)) +(1/2)e^(−(ω^2 /4)) ) I(α)=((π^(3/2) Γ(α+(1/2)))/(4Γ(α+1)))(ω^3 +2)e^(−(ω^2 /t)) (2):ln(1+x^2 )=∫_0 ^1 (x^2 /(1+tx^2 ))dt I(α)=2∫_0 ^1 ∫_0 ^1 ((x^4 e^(−x^2 ) cos(ωx))/((x^4 +1)^α (1+tx^2 )))dx dt cos(ωx)=Σ_(k=0) ^∞ (((−1)^k ω^(2k) x^(2k) )/((2k)!)) I(α)=2Σ_(k=0) ^∞ (((−1)^k ω^(2k) )/((2k)!))∫_0 ^1 ∫_0 ^∞ ((x^(4+2k) e^(−x^2 ) )/((x^4 +1)^α (1+tx^2 )))dx dt Let x^2 =y⇒:I(α)=Σ_(k=0) ^∞ (((−1)^k ω^(2k) )/((2k)!))∫_0 ^1 ∫_0 ^∞ ((y^(k+3/2) e^(−y) )/((y^2 +1)^α (1+ty)))dy dt ∫_0 ^∞ ((y^(k+3/2) e^(−y) )/((y^2 +1)^α ))dy=(1/2)Γ(k+(3/2)) _1 F_1 (k+(3/2);(1/2);(1/4)) I(α)=(1/2)Σ_(k=0) ^∞ (((−1)^k ω^(2k) Γ(k+(3/2)))/((2k)!))∫_0 ^1 _2 F_1 (1,k+(3/2);(1/2);−t)dt ∫_0 ^1 _2 F_1 (1,k+(3/2);(1/2);−t)dt=(2/(2k+1))(1− _2 F_1 (1,k+(3/2);(3/2);−1)) _2 F_(1 ) (1,k+(3/2);(3/2);−1)=(((√π)Γ(k+1))/(2^(2k+1) Γ(k+(3/2)))) I(α)=((√π)/2)Σ_(k=0) ^∞ (((−1)^k ω^(2k) Γ(k+1))/(2^(2k+1) (2k)!))(1−(1/2^(2k+1) )) Γ(k+1)=k!,Γ(k+(3/2))=(((√π)(2k+1)!)/(4^k k!)) I(α)=((√π)/2^(2α+1) )Σ_(k=0) ^∞ (((−1)^k ω^(2k) (2k)!)/((k!)^2 4^k )) _2 F_1 (α,α+(1/2);k+(3/2);(1/4)) (3): I(α)=((√π)/(2^α Γ(α)))Σ_(k=0) ^∞ (((−1)^k ω^(2k) )/((2k)!))∫_0 ^∞ ((t^(α−1) e^(−t) Γ(k+(3/2),t))/(Γ(k+(5/2))))dt
$$\mathrm{I}\:\mathrm{have}\:\mathrm{derived}\:\mathrm{three}\:\mathrm{resultsu} \\ $$$$\mathrm{bt}\:\mathrm{I}\:\mathrm{am}\:\mathrm{not}\:\mathrm{sure}\:\mathrm{which}\:\mathrm{one}\:\mathrm{isc} \\ $$$$\mathrm{correct}\:\left(\mathrm{you}\:\mathrm{can}\:\mathrm{verify}\:\mathrm{itr}\right. \\ $$$$\left.\mathrm{late}\right). \\ $$$$\left(\mathrm{1}\right):\Rightarrow{I}\left(\alpha\right)=\mathrm{2}\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{2}} \mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\alpha} }{e}^{−{x}^{\mathrm{2}} } \mathrm{cos}\left(\omega{x}\right){dx} \\ $$$$\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} {t}}{dt} \\ $$$$\Rightarrow{I}\left(\alpha\right)=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}\:} ^{\infty} \frac{{x}^{\mathrm{4}} }{\left({x}^{\mathrm{4}} +\mathrm{1}\right)\left(\mathrm{1}+{x}^{\mathrm{2}} {t}\right)}{e}^{−{x}^{\mathrm{2}} } \mathrm{cos}\left(\omega{x}\right) \\ $$$$\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{−\alpha} =\frac{\mathrm{1}}{\Gamma\left(\alpha\right)}\int_{\mathrm{0}} ^{\infty} {s}^{\alpha−\mathrm{1}} {e}^{−\left({x}^{\mathrm{4}} +\mathrm{1}\right){s}} {ds} \\ $$$$\Rightarrow{I}\left(\alpha\right)=\frac{\mathrm{2}}{\Gamma\left(\alpha\right)}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}\:} ^{\infty} {s}^{\alpha−\mathrm{1}} {e}^{−{s}} \int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{4}} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{1}+{sx}^{\mathrm{2}} \right)} }{\mathrm{1}+{x}^{\mathrm{2}} {t}}\mathrm{cos}\left(\omega{x}\right){dx}\:{ds}\:{dt} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{4}} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{1}+{sx}^{\mathrm{2}} \right)} \mathrm{cos}\left(\omega{x}\right){dx}=\frac{\sqrt{\pi}}{\mathrm{4}}\:\frac{\partial^{\mathrm{2}} }{\partial\lambda^{\mathrm{2}} }\left(\frac{{e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}\left(\mathrm{1}+{sx}^{\mathrm{2}} +\lambda\right)}} }{\:\sqrt{\mathrm{1}+{sx}^{\mathrm{2}} +\lambda}}\right)\mid_{\lambda=\mathrm{0}} \\ $$$$\Rightarrow{I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}\Gamma\left(\alpha\right)}\:\frac{\partial^{\mathrm{2}} }{\partial\lambda^{\mathrm{2}} }\left({e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}\left(\mathrm{1}+\lambda\right)}} \int_{\mathrm{0}} ^{\infty} \frac{{s}^{\alpha−\mathrm{1}} {e}^{−{s}} }{\:\sqrt{\mathrm{1}+{s}+\lambda}\left(\mathrm{1}+{t}\left(\mathrm{1}+{s}+\lambda\right)\right)}{ds}=\frac{\Gamma\left(\alpha\right)}{\left(\mathrm{1}+{t}\right)^{\alpha} \sqrt{\mathrm{1}+{t}}}\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\alpha,\frac{\mathrm{1}}{\mathrm{2}};\alpha+\frac{\mathrm{1}}{\mathrm{2}};\frac{{t}}{\mathrm{1}+{t}}\right)\right. \\ $$$${I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}}\:\frac{\partial^{\mathrm{2}} }{\partial\lambda^{\mathrm{2}} }\left(\frac{{e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}\left(\mathrm{1}−\lambda\right)}} }{\:\sqrt{\mathrm{1}+\lambda}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\left(\mathrm{1}+{t}\right)^{\alpha} }\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\alpha,\frac{\mathrm{1}}{\mathrm{2}};\alpha+\frac{\mathrm{1}}{\mathrm{2}};\frac{{t}}{\mathrm{1}+{t}}\right){dt}\right)\mid_{\lambda=\mathrm{0}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\alpha,\frac{\mathrm{1}}{\mathrm{2}};\alpha+\frac{\mathrm{1}}{\mathrm{2}};\frac{{t}}{\mathrm{1}+{t}}\right)}{\left(\mathrm{1}+{t}\right)^{\alpha} }{dt}=\frac{\sqrt{\pi}\Gamma\left(\alpha+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\Gamma\left(\alpha+\mathrm{1}\right)} \\ $$$$\Rightarrow{I}\left(\alpha\right)=\frac{\pi}{\mathrm{2}\Gamma\left(\alpha+\mathrm{1}\right)}\:\frac{\partial^{\mathrm{2}} }{\partial\lambda^{\mathrm{2}} }\left(\frac{{e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}\left(\mathrm{1}+\lambda\right)}} }{\:\sqrt{\mathrm{1}+\lambda}}\Gamma\left(\alpha+\frac{\mathrm{1}}{\mathrm{2}}\right)\right)\mid_{\lambda=\mathrm{0}} \\ $$$$\Rightarrow{I}\left(\alpha\right)=\frac{\pi^{\mathrm{3}/\mathrm{2}} \Gamma\left(\alpha+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{2}\Gamma\left(\alpha+\mathrm{1}\right)}\left(\frac{\omega^{\mathrm{4}} }{\mathrm{4}}{e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}}} +\frac{\mathrm{1}}{\mathrm{2}}{e}^{−\frac{\omega^{\mathrm{2}} }{\mathrm{4}}} \right) \\ $$$${I}\left(\alpha\right)=\frac{\pi^{\mathrm{3}/\mathrm{2}} \Gamma\left(\alpha+\frac{\mathrm{1}}{\mathrm{2}}\right)}{\mathrm{4}\Gamma\left(\alpha+\mathrm{1}\right)}\left(\omega^{\mathrm{3}} +\mathrm{2}\right){e}^{−\frac{\omega^{\mathrm{2}} }{{t}}} \\ $$$$\left(\mathrm{2}\right):\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+{tx}^{\mathrm{2}} }{dt} \\ $$$${I}\left(\alpha\right)=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{4}} {e}^{−{x}^{\mathrm{2}} } \mathrm{cos}\left(\omega{x}\right)}{\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\alpha} \left(\mathrm{1}+{tx}^{\mathrm{2}} \right)}{dx}\:{dt} \\ $$$$\mathrm{cos}\left(\omega{x}\right)=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} {x}^{\mathrm{2}{k}} }{\left(\mathrm{2}{k}\right)!} \\ $$$${I}\left(\alpha\right)=\mathrm{2}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} }{\left(\mathrm{2}{k}\right)!}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{4}+\mathrm{2}{k}} {e}^{−{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{4}} +\mathrm{1}\right)^{\alpha} \left(\mathrm{1}+{tx}^{\mathrm{2}} \right)}{dx}\:{dt} \\ $$$$\mathrm{Let}\:{x}^{\mathrm{2}} ={y}\Rightarrow:{I}\left(\alpha\right)=\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} }{\left(\mathrm{2}{k}\right)!}\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\infty} \frac{{y}^{{k}+\mathrm{3}/\mathrm{2}} {e}^{−{y}} }{\left({y}^{\mathrm{2}} +\mathrm{1}\right)^{\alpha} \left(\mathrm{1}+{ty}\right)}{dy}\:{dt} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{y}^{{k}+\mathrm{3}/\mathrm{2}} {e}^{−{y}} }{\left({y}^{\mathrm{2}} +\mathrm{1}\right)^{\alpha} }{dy}=\frac{\mathrm{1}}{\mathrm{2}}\Gamma\left({k}+\frac{\mathrm{3}}{\mathrm{2}}\right)\:_{\mathrm{1}} {F}_{\mathrm{1}} \left({k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$${I}\left(\alpha\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} \Gamma\left({k}+\frac{\mathrm{3}}{\mathrm{2}}\right)}{\left(\mathrm{2}{k}\right)!}\int_{\mathrm{0}} ^{\mathrm{1}} \:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\mathrm{1},{k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};−{t}\right){dt} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\mathrm{1},{k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{2}};−{t}\right){dt}=\frac{\mathrm{2}}{\mathrm{2}{k}+\mathrm{1}}\left(\mathrm{1}−\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\mathrm{1},{k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};−\mathrm{1}\right)\right) \\ $$$$\:_{\mathrm{2}} {F}_{\mathrm{1}\:} \left(\mathrm{1},{k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{3}}{\mathrm{2}};−\mathrm{1}\right)=\frac{\sqrt{\pi}\Gamma\left({k}+\mathrm{1}\right)}{\mathrm{2}^{\mathrm{2}{k}+\mathrm{1}} \:\Gamma\left({k}+\frac{\mathrm{3}}{\mathrm{2}}\right)} \\ $$$${I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} \Gamma\left({k}+\mathrm{1}\right)}{\mathrm{2}^{\mathrm{2}{k}+\mathrm{1}} \left(\mathrm{2}{k}\right)!}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}{k}+\mathrm{1}} }\right) \\ $$$$\Gamma\left({k}+\mathrm{1}\right)={k}!,\Gamma\left({k}+\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\sqrt{\pi}\left(\mathrm{2}{k}+\mathrm{1}\right)!}{\mathrm{4}^{{k}} {k}!} \\ $$$${I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}^{\mathrm{2}\alpha+\mathrm{1}} }\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} \left(\mathrm{2}{k}\right)!}{\left({k}!\right)^{\mathrm{2}} \mathrm{4}^{{k}} }\:_{\mathrm{2}} {F}_{\mathrm{1}} \left(\alpha,\alpha+\frac{\mathrm{1}}{\mathrm{2}};{k}+\frac{\mathrm{3}}{\mathrm{2}};\frac{\mathrm{1}}{\mathrm{4}}\right) \\ $$$$\left(\mathrm{3}\right): \\ $$$${I}\left(\alpha\right)=\frac{\sqrt{\pi}}{\mathrm{2}^{\alpha} \Gamma\left(\alpha\right)}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \omega^{\mathrm{2}{k}} }{\left(\mathrm{2}{k}\right)!}\int_{\mathrm{0}} ^{\infty} \frac{{t}^{\alpha−\mathrm{1}} {e}^{−{t}} \Gamma\left({k}+\frac{\mathrm{3}}{\mathrm{2}},{t}\right)}{\Gamma\left({k}+\frac{\mathrm{5}}{\mathrm{2}}\right)}{dt} \\ $$
Answered by SdC355 last updated on 04/May/25
I(α)=2∫_0 ^( ∞) ((z^2 ln(z^2 +1))/((z^2 +1)^α ))e^(−z^2 ) cos(ωz) dz cos^2 (θ)+sin^2 (θ)=1 → tan^2 (θ)+1=sec^2 (θ) z=tan(θ) (dz/dθ)=sec^2 (θ) → dz=sec^2 (θ) dθ 4∫_0 ^(π/2) tan^2 (θ)ln(sec(θ))e^(−tan^2 (θ)) cos(ω∙tan(θ))(cos(θ))^(2α−2) dθ
$${I}\left(\alpha\right)=\mathrm{2}\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{{z}^{\mathrm{2}} \mathrm{ln}\left({z}^{\mathrm{2}} +\mathrm{1}\right)}{\left({z}^{\mathrm{2}} +\mathrm{1}\right)^{\alpha} }{e}^{−{z}^{\mathrm{2}} } \mathrm{cos}\left(\omega{z}\right)\:\mathrm{d}{z} \\ $$$$\mathrm{cos}^{\mathrm{2}} \left(\theta\right)+\mathrm{sin}^{\mathrm{2}} \left(\theta\right)=\mathrm{1}\:\rightarrow\:\mathrm{tan}^{\mathrm{2}} \left(\theta\right)+\mathrm{1}=\mathrm{sec}^{\mathrm{2}} \left(\theta\right) \\ $$$${z}=\mathrm{tan}\left(\theta\right) \\ $$$$\frac{\mathrm{d}{z}}{\mathrm{d}\theta}=\mathrm{sec}^{\mathrm{2}} \left(\theta\right)\:\rightarrow\:\mathrm{d}{z}=\mathrm{sec}^{\mathrm{2}} \left(\theta\right)\:\mathrm{d}\theta \\ $$$$\mathrm{4}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{tan}^{\mathrm{2}} \left(\theta\right)\mathrm{ln}\left(\mathrm{sec}\left(\theta\right)\right){e}^{−\mathrm{tan}^{\mathrm{2}} \left(\theta\right)} \mathrm{cos}\left(\omega\centerdot\mathrm{tan}\left(\theta\right)\right)\left(\mathrm{cos}\left(\theta\right)\right)^{\mathrm{2}\alpha−\mathrm{2}} \mathrm{d}\theta \\ $$

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