Question

\[\int\sqrt{2ax - x^2} \text{ dx}\]

Answer 1

\[\text{ Let I }= \int\sqrt{2ax - x^2}\text{ dx}\]
\[ = \int\sqrt{a^2 + 2ax - x^2 - a^2}\text{ dx}\]
\[ = \int \sqrt{a^2 - \left( x^2 - 2ax + a^2 \right)}\text{ dx}\]
\[ = \int\sqrt{a^2 - \left( x - a \right)^2}\text{ dx}\]
\[ = \left( \frac{x - a}{2} \right) \sqrt{2ax - x^2} + \frac{a^2}{2} \sin^{- 1} \left( \frac{x - a}{a} \right) + C\]