How do I solve the integral $\int\frac{1-\sqrt{2x+3}}{1+\sqrt{2x+3}}dx$ with help substitution?

Question

How do I solve the integral $$\int\frac{1-\sqrt{2x+3}}{1+\sqrt{2x+3}}dx$$ with help substitution?

For example, if I set $t=\sqrt{2x+3}$, would that be a possible option? And if so, how would I go about it? Appreciate any hints and methods, since I'm new to this topic.

Answer 1

Yes, $ t = \sqrt{2x+3} $ is a good option.

${ t = \sqrt{2x+3} \Rightarrow t^2 = 2x+3 \Rightarrow x = {t^2-3\over2} \\ dx = t dt \\ \int{(1-t)t\over1+t}dt = - \int{{t^2-t}\over{t+1}}dt = -\int(t-2) dt - 2\int{dt\over t+1}}$.