Primitive of Power of Logarithm of x

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$\ds \int \ln^n x \rd x = x \ln^n x - n \int \ln^{n - 1} x \rd x + C$


With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$


\(\ds u\) \(=\) \(\ds \ln^n x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds n \ln^{n - 1} x \frac 1 x\) Derivative of $\ln x$, Derivative of Power, Chain Rule for Derivatives

and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds x\) Primitive of Constant


\(\ds \int \ln^n x \rd x\) \(=\) \(\ds x \ln^n x - \int x \paren {n \ln^{n - 1} x \frac 1 x} \rd x + C\) Integration by Parts
\(\ds \) \(=\) \(\ds x \ln^n x - n \int \ln^{n - 1} x \rd x + C\) Primitive of Constant Multiple of Function