Primitive of Reciprocal of Logarithm of x has no Solution in Elementary Functions

Theorem

The primitive:

$\ds \int \frac {\d x} {\ln x}$

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

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Primitive of $\dfrac {\d x} {\ln x}$

Historical Note

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

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.29$: Liouville ($\text {1809}$ – $\text {1882}$)