Question

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

Answer 1

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