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

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Theorem

The primitive:

$\ds \int \frac {e^{a x} \rd x} x$

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


Proof



Also see


Historical Note

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


Sources