Question:
$\sqrt{1-4 x-x^{2}}$
Solution:
Let $I=\int \sqrt{1-4 x-x^{2}} d x$
$=\int \sqrt{1-\left(x^{2}+4 x+4-4\right)} d x$
$=\int \sqrt{1+4-(x+2)^{2}} d x$
$=\int \sqrt{(\sqrt{5})^{2}-(x+2)^{2}} d x$
It is known that, $\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1} \frac{x}{a}+\mathrm{C}$
$\therefore I=\frac{(x+2)}{2} \sqrt{1-4 x-x^{2}}+\frac{5}{2} \sin ^{-1}\left(\frac{x+2}{\sqrt{5}}\right)+\mathrm{C}$