Question

Show that $\int \frac{dx}{(x+2)\sqrt{(x-1)}}=\frac{2}{\sqrt{\left(3\right)}}{\tan }^{-1}\sqrt{\left(\frac{x-1}{3}\right)}$.

Answer 1

Let $I=\int \frac{1}{(x+2)\sqrt{x-1}}dx$
Put $t=\sqrt{x-1}\Rightarrow t^2=x-1$$

$\Rightarrow dt=\frac{1}{2\sqrt{x-1}}dx$
$I=\int \frac{2}{t^2+3}dt$

$=\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{t}{\sqrt{3}}\right)+C$

$I=\frac{2}{\sqrt{3}}{\tan }^{-1}\left(\frac{\sqrt{x-1}}{\sqrt{3}}\right)+C$