Primitive of Power of Logarithm of x

Theorem

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

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: