Question

$\int \frac{1}{\left(x-1\right)\sqrt{x+2}}dx$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ I=\int \frac{dx}{\left(x-1\right)\sqrt{x+2}} \\ \mathrm{Putting}x+2=t^2 \\ \Rightarrow dx=2tdt \\ \therefore I=\int \frac{2tdt}{\left(t^2-2-1\right)t} \\ =\int \frac{2dt}{t^2-{\left(\sqrt{3}\right)}^2} \\ =2\times \frac{1}{2\sqrt{3}}\log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+\mathrm{C} \\ =\frac{1}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+\mathrm{C} \end{gathered}$$