Question

Solve:

$\int \frac{1}{\sqrt{x+1}+\sqrt{x}}dx$

Answer 1

$\int \frac{1}{\sqrt{x+1}+\sqrt{x}}=\int \frac{\sqrt{x+1}-\sqrt{x}}{(\sqrt{x+1}{)}^2-(\sqrt{x}{)}^2}$

$=\int \frac{\sqrt{x+1}-\sqrt{x}}{x+1-x}=\int \sqrt{x+1}-\int \sqrt{x}$

$=\frac{(x+1{)}^{3/2}}{3/2}-\frac{x^{3/2}}{3/2}$

$=\frac{2}{3}\left[\right(x+1{)}^{3/2}-x^{3/2}]$