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

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

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Also see

• Primitive of $\dfrac {e^{a x} } x$

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

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor: Example $2$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)