Primitive of Reciprocal of x by Logarithm of x

Theorem

$\ds \int \frac {\d x} {x \ln x} = \ln \size {\ln x} + C$

Proof

\(\ds z\)

\(=\)

\(\ds \ln x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d z} {\d x}\)

\(=\)

\(\ds \frac 1 x\)

Derivative of Natural Logarithm

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\d x} {x \ln x}\)

\(=\)

\(\ds \ln \size {\ln x} + C\)

Primitive of Function under its Derivative

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\ln x$: $14.532$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals