Question

Evaluate :

$\int \frac{x}{(x-3)\sqrt{x+1}}dx$

• A. $2\sqrt{x+1}+\frac{3}{2}ln\left|\frac{\sqrt{x-1}-2}{\sqrt{x-1}+2}\right|+c$
• B. $2\sqrt{x-1}+\frac{3}{2}ln\left|\frac{\sqrt{x+1}-2}{\sqrt{x+1}+2}\right|+c$
• C. $2\sqrt{x+1}+\frac{3}{2}ln\left|\frac{\sqrt{x+1}-2}{\sqrt{x+1}+2}\right|+c$
• D. $2\sqrt{x+1}+\frac{2}{3}ln\left|\frac{\sqrt{x-1}-2}{\sqrt{x+1}+2}\right|+c$

Answer 1

The correct option is C $2\sqrt{x+1}+\frac{3}{2}ln\left|\frac{\sqrt{x+1}-2}{\sqrt{x+1}+2}\right|+c$

Let $I=\int \frac{x}{(x+3)\sqrt{x+1}}dx$
Put $t=\sqrt{x+1}\Rightarrow dt=\frac{1}{2\sqrt{x+1}}dx$
$I=\int \frac{2(t^2-1)}{t^2-4}dt=2\int (-\frac{3}{4(t+2)}+\frac{3}{4(t-2)}+1)dt$
$=-\frac{3}{2}\int \frac{1}{t+2}dt+\frac{3}{2}\int \frac{1}{t-2}dt+2\int dt$
$=-\frac{3}{2}\log (t+2)+\frac{3}{2}\log (t-2)+2t$
$=-\frac{3}{2}\log (\sqrt{x+1}+2)+\frac{3}{2}\log (\sqrt{x+1}-2)+2\sqrt{x+1}$
$=\frac{3}{2}\log \left(\frac{\sqrt{x+1}-2}{\sqrt{x+1}+2}\right)+2\sqrt{x+1}$