Question

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

Answer 1

$$\text{ We have, }$$
$$I=\int \frac{dx}{(x-1)\sqrt{2x+3}}$$
$$\text{ Putting 2x}+3=t^2$$
$$\Rightarrow x=\frac{t^2-3}{2}$$
$$\text{ Diff both sides}$$
$$dx=t\text{ dt}$$
$$\therefore I=\int \frac{tdt}{[\frac{t^2-3}{2}-1]t}$$
$$=\int \frac{2\text{ dt }}{t^2-3-2}$$
$$=\frac{2\text{ dt}}{t^2-5}$$
$$=2\int \frac{dt}{t^2-{\left(\sqrt{5}\right)}^2}$$
$$=2\times \frac{1}{2\sqrt{5}}\text{ log }\left|\frac{t-\sqrt{5}}{t+\sqrt{5}}\right|+C$$
$$=\frac{1}{\sqrt{5}}\text{ log }\left|\frac{\sqrt{\text{ 2x + 3}}-\sqrt{5}}{\sqrt{2x+3}+\sqrt{5}}\right|+C$$