Question

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

• A. ${\sin }^{-1}\frac{x+2}{\left(5\right)}$
• B. ${\sin }^{-1}\frac{x-2}{\sqrt{\left(5\right)}}$
• C. ${\cos }^{-1}\frac{x+2}{\sqrt{\left(5\right)}}$
• D. ${\sin }^{-1}\frac{x+2}{\sqrt{\left(5\right)}}$

Answer 1

The correct option is D ${\sin }^{-1}\frac{x+2}{\sqrt{\left(5\right)}}$

Let $I=\int \frac{1}{\sqrt{(1-4x-x^2)}}dx=\int \frac{dx}{\sqrt{[5-{(x+2)}^2]}}$
Using $\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\frac{x}{a}$
We get
$I={\sin }^{-1}\frac{x+2}{\sqrt{5}}$
Hence, option 'D' is correct.