Question

$\int \sqrt{9-x^2}dx$

Answer 1

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