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