Question

\[\int\sqrt{9 - x^2}\text{ dx}\]

Answer 1

\[\int \sqrt{9 - x^2} \text{ dx}\]
\[ = \int \sqrt{3^2 - x^2} \text{ dx}\]
\[ = \frac{x}{2}\sqrt{3^2 - x^2} + \frac{3^2}{2} \sin^{- 1} \left( \frac{x}{3} \right) + C \left[ \because \int\sqrt{a^2 - x^2} \text{ dx } = \frac{1}{2}x\sqrt{a^2 - x^2} + \frac{1}{2} a^2 \text{ sin}^{- 1} \left( \frac{x}{a} \right) + C \right]\]
\[ = \frac{x}{2}\sqrt{9 - x^2} + \frac{9}{2} \sin^{- 1} \left( \frac{x}{3} \right) + C\]