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Prove-or-disprove-0-a-x-2-e-x-2-dx-1-e-a-2-a-3-1-3-2a-5-1-3-5-2a-7-1-3-5-7-2a-9-1-3-5-7-9-ad-inf-




Question Number 138608 by Dwaipayan Shikari last updated on 15/Apr/21
Prove or disprove  ∫_0 ^a x^2 e^(−x^2 ) dx=(1/e^a^2  )((a^3 /(1.3))+((2a^5 )/(1.3.5))+((2a^7 )/(1.3.5.7))+((2a^9 )/(1.3.5.7.9))+ad inf..)
$${Prove}\:{or}\:{disprove} \\ $$$$\int_{\mathrm{0}} ^{{a}} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } {dx}=\frac{\mathrm{1}}{{e}^{{a}^{\mathrm{2}} } }\left(\frac{{a}^{\mathrm{3}} }{\mathrm{1}.\mathrm{3}}+\frac{\mathrm{2}{a}^{\mathrm{5}} }{\mathrm{1}.\mathrm{3}.\mathrm{5}}+\frac{\mathrm{2}{a}^{\mathrm{7}} }{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}}+\frac{\mathrm{2}{a}^{\mathrm{9}} }{\mathrm{1}.\mathrm{3}.\mathrm{5}.\mathrm{7}.\mathrm{9}}+{ad}\:{inf}..\right) \\ $$

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