Question

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

Answer 1

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