Primitive of Function of Natural Logarithm

Theorem

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

where $u = \ln x$.

Proof

\(\ds u\)

\(=\)

\(\ds \ln x\)

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \frac 1 x\)

Derivative of Natural Logarithm Function

\(\ds \leadsto \ \ \)

\(\ds \int \map F {\ln x} \rd x\)

\(=\)

\(\ds \int \map F u x \rd u\)

Primitive of Composite Function

\(\ds \)

\(=\)

\(\ds \int \map F u e^u \rd u\)

Definition of Exponential Function

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Important Transformations: $14.56$