Question

${\int }_0^1\frac{dx}{\sqrt{1+x}-\sqrt{x}}dx$

Answer 1

$I={\int }_0^1\frac{dx}{\sqrt{1+x}-\sqrt{x}}$

$I={\int }_0^1(\sqrt{1+x}+\sqrt{x})dx$

$I={\left[\frac{2}{3}\right(1+x{)}^{\frac{3}{2}}+\frac{2}{3}\left(x{)}^{\frac{3}{2}}\right)]}_0^1$

$I=\frac{2}{3}{\left[\right(1+x{)}^{\frac{3}{2}}+\left(x{)}^{\frac{3}{2}}\right)]}_0^1$

$I=\frac{2}{3}\left[\right(1+1{)}^{\frac{3}{2}}+\left(1{)}^{\frac{3}{2}}\right)-\left[\right(1+0{)}^{3/2}-0\left]\right]$

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

$I=\frac{2^{\frac{5}{2}}}{3}$

Hence, this is the answer.