Question:
Evaluate the following integral:
$\int \frac{1}{(x-1) \sqrt{x+2}} d x$
Solution:
assume $x+2=t^{2}$
$\mathrm{d} \mathrm{x}=2 \mathrm{tdt}$
$\int \frac{2 d t}{\left(t^{2}-3\right)}$
Using identity $\int \frac{d z}{(z)^{2}-1}=\frac{1}{2} \log \left|\frac{z-1}{z+1}\right|+c$
$\frac{1}{\sqrt{3}} \log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+c$
$\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{(x+2)}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+c$