Primitive of Function of Natural Logarithm

Theorem

$\ds \int \map F {\ln x} \rd x = \int \map F u e^u \rd u$

where $u = \ln x$.

$\ds u$

$=$

$\ds \ln x$

$\ds \leadsto \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds \frac 1 x$

$\ds \leadsto \ \ $

$\ds \int \map F {\ln x} \rd x$

$=$

$\ds \int \map F u x \rd u$

$\ds $

$=$

$\ds \int \map F u e^u \rd u$

Definition of Exponential Function

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.56$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: Important Transformations: $16.56.$