Question

Integrate the function $\sqrt{1-4x-x^2}$

Answer 1

Let $I=\int \sqrt{1-4x-x^2}dx$
$=\int \sqrt{1-(x^2+4x+4-4)}dx$
$=\int \sqrt{1+4-(x+2{)}^2}dx$
$=\int \sqrt{(\sqrt{5}{)}^2-(x+2{)}^2}dx$
We know that, $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}+C$
$\therefore I=\frac{(x+2)}{2}\sqrt{1-4x-x^2}+\frac{5}{2}{\sin }^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+C$