Question

Evaluate $\int \frac{dx}{\sqrt{x+1}-\sqrt{x}}$.

Answer 1

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

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

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

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

$=\frac{{(x+1)}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c$

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