Question

The value of integral $\int \frac{dx}{(x+2)\sqrt{x+1}}$ is (where $C$ is integration constant)

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

Answer 1

The correct option is A $2{\tan }^{-1}\left(\sqrt{x+1}\right)+C$

$I=\int \frac{dx}{(x+2)\sqrt{x+1}}$
Put $x+1=t^2\Rightarrow dx=2tdt$
$\Rightarrow I=\int \frac{2tdt}{(t^2+1)t}=2{\tan }^{-1}t+C$
$=2{\tan }^{-1}\left(\sqrt{x+1}\right)+C$