Find: ∫ (x+2)/(√(9 x-x^2)) d x

$\displaystyle I=\int \dfrac{x+2}{\sqrt{9x-x^{2}}}\, dx$

Let $\displaystyle x+2=A(9-2x)+B$

$\displaystyle \Rightarrow A=-\dfrac{1}{2}, B=\dfrac{13}{2}$

$\displaystyle I=-\dfrac{1}{2}\int \dfrac{9-2x}{\sqrt{9x-x^{2}}}\, dx+\dfrac{13}{2}\int \dfrac{1}{\sqrt{9x-x^{2}}}\, dx$

$\displaystyle =-\dfrac{1}{2}\left(2\sqrt{9x-x^{2}}\right)+\dfrac{13}{2}\int \dfrac{1}{\sqrt{\left(\dfrac{9}{2}\right)^{2}-\left(x-\dfrac{9}{2}\right)^{2}}}\, dx$

$\displaystyle =-\sqrt{9x-x^{2}}+\dfrac{13}{2} \sin^{-1}\left(\dfrac{2x-9}{9}\right)+C$