Primitive of Power of x by Power of Logarithm of x

Theorem

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

where $m \ne -1$.

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\ln^n x} x$ for the case where $m = -1$.
 * Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x