Question

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

Answer 1

Let $I=\int \sqrt{1-4x-x^2}dx$

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

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

=$\int\sqrt{(\sqrt5)^2-(x+2)^2}dx$

It is known that,$\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C$

∴$I=\frac{(x+2)}{2}\sqrt{1-4x-x^2}+\frac{5}{2}\sin^{-1}\bigg(\frac{x+2}{\sqrt 5}\bigg)+C$