Primitive of Root of 2 a x minus x squared

Theorem

 * $\ds \int \sqrt {2 a x - x^2} \rd x = \frac {\paren {x - a} } 2 \sqrt {2 a x - x^2} + \frac {a^2} 2 \arcsin \frac {x - a} a + C$

where $C$ is an arbitrary constant.

Proof
Let $u := x - a$.

Then:
 * $\dfrac {\d u} {\d x} = 1$

and:
 * $x = u + a$

Then:

and we have: