Answer
$- \sqrt{4-x^2} +5\sin^{-1}(x/2)+c.$
Work Step by Step
We have
$$\int \frac{(x+5)dx}{\sqrt{4-x^2}}=\int \frac{xdx}{\sqrt{4-x^2}}+\int \frac{5dx}{\sqrt{4-x^2}}\\
=-\frac{1}{2}\int \frac{-2xdx}{\sqrt{4-x^2}}+5\int \frac{d(x/2)}{\sqrt{1-(x/2)^2}}\\
=-\frac{1}{2} \frac{\sqrt{4-x^2}}{1/2}+5\sin^{-1}(x/2)+c\\
=- \sqrt{4-x^2} +5\sin^{-1}(x/2)+c.$$
