Question

Evaluate : $\int \frac{1}{(x+1)\sqrt{3+2x-x}}dx$

Answer 1

$\begin{array}{c}\int \frac{dx}{\left(x+1\right)\sqrt{3+2x-x}}\\ =\int \frac{dx}{\left(x+1\right)\sqrt{x+3}}\\ x+3=t^2\frac{dx}{ddt}=2t\\ =\int \frac{ztdt}{\left(t^2-2\right)t}\\ =2\int \frac{dt}{t^2-2}\\ =2\times \frac{1}{2\sqrt{2}}In\left|\frac{t-\sqrt{2}}{t+\sqrt{2}}\right|+C\\ =\frac{1}{\sqrt{2}}In\left|\frac{\sqrt{x+3}-\sqrt{2}}{\sqrt{x+3}+\sqrt{2}}\right|+C\end{array}$