Question

Evaluate$\int \frac{1}{(x-3)\sqrt{x+1}}dx.$
The ans is $=\frac{1}{2}log\left|\frac{\sqrt{x+1}-k}{\sqrt{x+1}+k}\right|+C$
Find k?

Answer 1

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