Primitive of Function of Natural Logarithm
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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: $14.56$