integrate dx/√x+√x-2

Dear student,
$$\begin{gathered} I=\int \frac{dx}{\sqrt{x}+\sqrt{x-2}} \\ multiply and divide by \sqrt{x}-\sqrt{x-2} \\ =\int \frac{{\displaystyle \sqrt{x}-\sqrt{x-2}}}{{\displaystyle \sqrt{x}-\sqrt{x-2}}}\times \frac{{\displaystyle dx}}{{\displaystyle \sqrt{x}+\sqrt{x-2}}} \\ =\int \frac{{\displaystyle \sqrt{x}-\sqrt{x-2}}}{{\displaystyle x-x+2}}dx \\ =\frac{1}{2}\int \left(\sqrt{x}-\sqrt{x-2}\right)dx \\ =\frac{{\displaystyle 1}}{{\displaystyle 2}}\left(\frac{2}{3}{x}^{\frac{3}{2}}-\frac{{\displaystyle 2}}{{\displaystyle 3}}{\left(x-2\right)}^{\frac{3}{2}}\right)+c \\ I =\frac{{\displaystyle 1}}{{\displaystyle 3}}\left(x^{\frac{3}{2}}-{\left(x-2\right)}^{\frac{3}{2}}\right) +c answer \end{gathered}$$
Hope this information will clear your doubts. If you have any more doubts just ask here on the forum and our experts will try to help out as soon as possible.
Regards