Primitive of Power of x by Power of Logarithm of x

Theorem

 * $\displaystyle \int x^m \ln^n x \ \mathrm d x = \frac {x^{m + 1} \ln^n x} {m + 1} - \frac n {m + 1} \int x^m \ln^{n - 1} x \ \mathrm d x + C$

where $m \ne -1$.

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

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {\ln^n x} x$ for the case where $m = -1$.