Question

Evaluate the following integral:
$\int \frac{dx}{\sqrt{x+1}-\sqrt{x}}$

Answer 1

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

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

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

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

By using $\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$

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