Can anyone show if the following integral can be evaluated in closed form? \begin{equation} \int_{-1}^1 e^{ax^2+bx+c\sqrt{1-x^2}}dx \end{equation} The variable $x$ can be replaced by $\cos{\theta}$, with corresponding change of the interval of integration.
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are you looking to show that it CAN be integrated? or to find a closed form for the integral. because the former probably easier in this case. – pancini Aug 18 '15 at 08:40
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Closed form seems unlikely.. – tired Aug 18 '15 at 08:44
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Do you think that a closed form could exist for the antiderivative ? – Claude Leibovici Aug 18 '15 at 08:44
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@ClaudeLeibovici not at all, the combination of the $ a x^2+c \sqrt{1-x^2}$ seems not very promising – tired Aug 18 '15 at 08:49
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@ClaudeLeibovici I don't know if it exists. I have modified the question to reflect that. – khalatnikov Aug 18 '15 at 08:49
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@tired. I bet that not very promising is an understatement ! Cheers :-) – Claude Leibovici Aug 18 '15 at 08:51
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@ClaudeLeibovici True :) But it may be possible to make some nice asymptotics in some of the parameters.... – tired Aug 18 '15 at 09:04
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@tired. I agree, for sure. Taylor expansion will work. – Claude Leibovici Aug 18 '15 at 09:18
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@tired, the OP didn't mention asymptotics at all, so the tag isn't relevant. I'm rolling back the edit. – Antonio Vargas Aug 18 '15 at 11:06
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@AntonioVargas but it is the best what u can do with this integral. Maybe he should give us a hint if he is interested or not – tired Aug 18 '15 at 11:08