Question

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

• A. $2\sqrt{x+2}+\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x-2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C$
• B. $2\sqrt{x+2}+\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C$
• C. $2\sqrt{x+2}-\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x-2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C$
• D. $2\sqrt{x+2}+\frac{2}{\sqrt{2}}\log \left|\frac{\sqrt{x-2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|$

Answer 1

The correct option is B $2\sqrt{x+2}+\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C$

$I=\int \frac{x+1}{(x-1)\sqrt{x+2}}dx$
Let $x+2=t^2$
or $dx=2tdt$
$\therefore I=\int \frac{t^2-1}{(t^2-3)t}2tdt$
$=2\int \frac{t^2-3+2}{(t^2-3)}dt$
$=2\int (1+\frac{2}{(t^2-3)})dt$
$=2t+\frac{2}{\sqrt{3}}\log \left|\frac{t-\sqrt{3}}{t+\sqrt{3}}\right|+C$
$=2\sqrt{x+2}+\frac{2}{\sqrt{3}}\log \left|\frac{\sqrt{x+2}-\sqrt{3}}{\sqrt{x+2}+\sqrt{3}}\right|+C$