Question

Evaluate: $\int \frac{x+2}{\sqrt{4x-x^2}}dx$

Answer 1

$\int \frac{x+2}{\sqrt{4x-x^2}}dx=\int \frac{(x-2)+4}{\sqrt{4-(x-2{)}^2}}dx=\int \frac{(x-2)}{\sqrt{4-(x-2{)}^2}}dx+4\int \frac{1}{\sqrt{4-(x-2{)}^2}}dx=\sqrt{4x-x^2}+4{\text{sin}}^{-1}(\frac{x-2}{2})+C$