What is the integration of \[ \int {\sqrt {2ax – {x^2}} } dx\] ~~~~~[NCERT Exemp. Q. 20,Page 164 ]

Let \[ I = \int {\sqrt {2ax – {x^2}} } dx = \int {\sqrt { – \left( {{x^2} – 2ax} \right)dx} } \]


\[ = \int {\sqrt { – \left( {{x^2} – 2ax + {a^2} – {a^2}} \right)} } dx = \int {\sqrt { – {{(x – a)}^2} – {a^2}} } dx\]


\[ = \int {\sqrt {{a^2} – {{(x – a)}^2}} } dx\]


\[ = \frac{{x – a}}{2}\sqrt {{a^2} – {{(x – a)}^2}} + \frac{{{a^2}}}{2}{\sin ^{ – 1}}\left( {\frac{{x – a}}{a}} \right) + C\]


\[ = \frac{{x – a}}{2}\sqrt {2ax – {x^2}} + \frac{{{a^2}}}{2}{\sin ^{ – 1}}\left( {\frac{{x – a}}{a}} \right) + C\]

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