Question

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

Answer 1

${\int }_0^1\frac{1}{\sqrt{1+x}-\sqrt{x}}dx{\int }_0^1\frac{1}{\sqrt{1+x}-\sqrt{x}}\times \frac{\sqrt{1+x}+\sqrt{x}}{\sqrt{1+x}+\sqrt{x}}dx{\int }_0^1\frac{\sqrt{1+x}+\sqrt{x}}{1+x-x}dx{\int }_0^1\sqrt{1+x}dx+{\int }_0^1\sqrt{x}dx\frac{(1+x{)}^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\frac{2}{3}\left(\right(1+x{)}^{\frac{3}{2}}+x^{\frac{3}{2}})+c$

Put limits and solve

=$\frac{4}{3}$$\sqrt{2}$