Question

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

Answer 1

$\int \frac{dx}{\left(\sqrt{2x+3}+\sqrt{2x}-3\right)}$
Rationalise the denominator
$$\begin{gathered} =\int \frac{\left(\sqrt{2x+3}-\sqrt{2x-3}\right)}{\left(\sqrt{2x+3}+\sqrt{2x-3}\right)\left(\sqrt{2x+3}-\sqrt{2x-3}\right)}dx \\ =\int \frac{\left(\sqrt{2x+3}-\sqrt{2x-3}\right)}{\left(2x+3\right)-\left(2x-3\right)}dx \\ =\frac{1}{6}\int {\left(2x+3\right)}^{\frac{1}{2}}dx-\frac{1}{6}\int {\left(2x-3\right)}^{\frac{1}{2}}dx \\ =\frac{1}{6}\left[\frac{{\left(2x+3\right)}^{{\displaystyle \frac{1}{2}}+1}}{2\left({\displaystyle \frac{1}{2}}+1\right)}\right]-\frac{1}{6}\left[\frac{{\left(2x-3\right)}^{{\displaystyle \frac{1}{2}}+1}}{2\left({\displaystyle \frac{1}{2}}+1\right)}\right]+\mathrm{C} \\ =\frac{1}{18}\left\{{\left(2x+3\right)}^{\frac{3}{2}}-{\left(2x-3\right)}^{\frac{3}{2}}\right\}+\mathrm{C} \end{gathered}$$