Primitive of x by Square of Logarithm of x

Theorem

 * $\ds \int x \ln^2 x \rd x = \dfrac {x^2 \ln^2 x} 2 - \dfrac {x^2 \ln x} 2 + \dfrac {x^2} 4 + C$

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: