prove that: ∫_(−(1/2)) ^∞ e^(−(4x^6 +12x^5 +15x^4 +10x^3 +4x^2 +x)) dx =((e)^(1/8) /3)[((Γ((1/6))^((−1)/2) )/(2(2)^(1/3) ))1F2(_(1/3,2/3) ^(1/6) ∣((−1)/(69/2)) ) +((Γ(5/6))/(128(4)^(1/3) ))1F2(_(4/3,5/3) ^(5/6) ∣((−1)/(69/2))) −((√π)/(16))12(


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Question Number 99120 by M±th+et+s last updated on 18/Jun/20

$p r o v e t h a t :$ $\int_{- \frac{1}{2}}^{\infty} e^{- (4 x^{6} + 12 x^{5} + 15 x^{4} + 10 x^{3} + 4 x^{2} + x)} d x$ $= \frac{\sqrt[8]{e}}{3} [\frac{Γ {(\frac{1}{6})}^{\frac{- 1}{2}}}{2 \sqrt[3]{2}} 1 F 2 (_{1 / 3, 2 / 3}^{1 / 6} ∣ \frac{- 1}{69 / 2}) + \frac{Γ (5 / 6)}{128 \sqrt[3]{4}} 1 F 2 (_{4 / 3, 5 / 3}^{5 / 6} ∣ \frac{- 1}{69 / 2}) - \frac{\sqrt{π}}{16} 12 (_{2 / 3, 4 / 3}^{1 / 2} ∣ \frac{- 1}{69 / 2})$
Commented by M±th+et+s last updated on 18/Jun/20

$I = \frac{\sqrt[8]{e} . π \sqrt{π}}{2 \sqrt[3]{4} \sqrt[6]{3}} [{(Ai (\frac{- 1}{8 \sqrt[3]{6}}))}^{2} + {(Bi (\frac{- 1}{8 \sqrt[3]{6}}))}^{2}]$
Commented by M±th+et+s last updated on 18/Jun/20

$Ai (z) : A i r y f u n c t i o n$ $Bi (z) : A i r y Bi f u n c t i o n$ $1 F 2 : i s h y p e r g e o m e t r i c f u n c t i o n$ $Γ (s) : g a m m a f u n c t i o n$
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