$\int {\sqrt {2ax - {x^2}} } dx$

Let $I = \int {\sqrt {2ax - {x^2}} } dx = \int {\sqrt { - \left( {{x^2} - 2ax} \right)dx} }$

$= \int {\sqrt { - \left( {{x^2} - 2ax + {a^2} - {a^2}} \right)} } dx = \int {\sqrt { - {{(x - a)}^2} - {a^2}} } dx$

$= \int {\sqrt {{a^2} - {{(x - a)}^2}} } dx$

$= \frac{{x - a}}{2}\sqrt {{a^2} - {{(x - a)}^2}} + \frac{{{a^2}}}{2}{\sin ^{ - 1}}\left( {\frac{{x - a}}{a}} \right) + C$

$= \frac{{x - a}}{2}\sqrt {2ax - {x^2}} + \frac{{{a^2}}}{2}{\sin ^{ - 1}}\left( {\frac{{x - a}}{a}} \right) + C$