Question

Evaluate :
$\int \frac{1}{(x+3)\sqrt{x+2}}dx$, on $x\in I\subset (-2,\infty )$

Answer 1

Let $I=\int \frac{1}{(x+3)\sqrt{x+2}}dx$

Substitute $x+2=t^2\Rightarrow dx=2tdt$

So, $I=\int \frac{2tdt}{(t^2-2+3)t}=2\int \frac{dt}{1+t^2}$

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