Primitive of exp x over x has no Solution in Elementary Functions

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Theorem

The primitive:

$\displaystyle \int \frac {e^{a x} \, \mathrm d x} x$

cannot be expressed in terms of a finite number of elementary functions.


Proof


Also see


Historical Note

The proof that $\displaystyle \int \dfrac {e^x \, \mathrm d x} x$ cannot be expressed with a finite number of elementary functions was first proved by Joseph Liouville.


Sources