Question

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

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ I=\int \frac{x+1}{\left(x-1\right)\sqrt{x+2}}dx \\ \mathrm{Putting}x+2=t^2 \\ \Rightarrow x=t^2-2 \\ \mathrm{Diff}\mathrm{both}\mathrm{sides} \\ dx=2tdt \\ I=\int \frac{\left(t^2-2+1\right)2tdt}{\left(t^2-2-1\right)t} \\ =2\int \left(\frac{t^2-1}{t^2-3}\right)dt \\ =2\int \left(\frac{t^2-3+2}{t^2-3}\right)dt \\ =2\int \left(\frac{t^2-3}{t^2-3}\right)dt+4\int \frac{dt}{t^2-3} \\ =2\int dt+4\int \frac{dt}{t^2-{\left(\sqrt{3}\right)}^2} \\ =2t+4\times \frac{1}{2\sqrt{3}}\log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+\mathrm{C} \\ =2\sqrt{x+2}+\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+\mathrm{C} \end{gathered}$$