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As a follow-up to a question on evaluating the definite integral $\int_0^\infty \mathrm{e}^{-x^n}\,\mathrm{d}x$, I wish to know if there is a general analytic solution to the related integral where $-x^n$ is replaced by a polynomial of arbitrary degree, namely $$ \int_0^\infty \mathrm{e}^{\sum_ia_ix^{n_i}}\mathrm{d}x $$ for $n_i\in\mathbb{Z}$ and where the individual coefficients $a_i\in\mathbb{R}$ can be positive or negative, but in a manner such that the argument of the exponent (i.e,. the polynomial) is negative.

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    Well, if $p(x)$ is a polynomial, then so is $p(x-a)$ for any $a$, so $\int_0^\infty exp(p(x-a))dx=\int_a^\infty exp(p(x))dx$ is asking for the INDEFINITE integral of $\exp(p(x))$.2012-02-18

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