Question

Integrate the following:
$\int \frac{x+3}{\sqrt{5-x^2-4x}}$

Answer 1

$\int \frac{x+3}{\sqrt{5-x^2-4x}}dx$

Let $5-x^2-4x=t^2⟹(x+2)dx=-tdt$

Now $\int \frac{x+2}{\sqrt{5-x^2-4x}}dx$$+\int \frac{1}{\sqrt{5-x^2-4x}}dx$

$=\int \frac{-t}{t}dx$$+\int \frac{1}{\sqrt{3^2-(x+2{)}^2}}dx$

$=-t+{\sin }^{-1}\frac{x+2}{3}+c$

$=-\sqrt{5-x^2-4x}++{\sin }^{-1}\frac{x+2}{3}+c$