Primitive of Power of Logarithm of x over x
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Theorem
- $\ds \int \frac {\ln^n x} x \rd x = \frac {\ln^{n + 1} x} {n + 1} + C$
Proof
\(\ds z\) | \(=\) | \(\ds \ln x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds \frac 1 x\) | Derivative of Natural Logarithm | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\ln^n x} x \rd x\) | \(=\) | \(\ds \int z^n \rd z\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {z^{n + 1} } {n + 1} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\ln^{n + 1} x} {n + 1} + C\) | substituting for $z$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\ln x$: $14.531$