Primitive of x by Logarithm of x/Proof 1

Theorem

 * $\displaystyle \int x \ln x \ \mathrm d x = \frac {x^2} 2 \left({\ln x - \frac 1 2}\right) + C$

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: