Answer
\[\frac{2}{3}\left[(x+1)^{\frac{3}{2}}-x^{\frac{3}{2}}\right]+C\]
Work Step by Step
Let \[I=\int\frac{1}{\sqrt{x+1}+\sqrt{x}}dx\]
\[I=\int\frac{1}{\sqrt{x+1}+\sqrt{x}}\times\left(\frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x+1}-\sqrt{x}}\right)dx\]
\[I=\int\left[\frac{\sqrt{x+1}-\sqrt{x}}{x+1-x}\right]dx\]
\[I=\int[\sqrt{x+1}-\sqrt{x}]dx\]
\[I=\frac{2(x+1)^{\frac{3}{2}}}{3}-\frac{2x^{\frac{3}{2}}}{3}+C\]
Hence \[\;\;I=\frac{2(x+1)^{\frac{3}{2}}}{3}-\frac{2x^{\frac{3}{2}}}{3}+C\]
Where $C$ is constant of integration
\[I=\frac{2}{3}\left[(x+1)^{\frac{3}{2}}-x^{\frac{3}{2}}\right]+C\]
Hence, \[I=\frac{2}{3}\left[(x+1)^{\frac{3}{2}}-x^{\frac{3}{2}}\right]+C.\]
