Question

Integrate $\int \frac{dx}{\sqrt{x+1}+\sqrt{x+2}}dx$

Answer 1

$\int \frac{dx}{\sqrt{x+1}+\sqrt{x+2}}$

$=\int \frac{\left(\sqrt{x+1}-\sqrt{x+2}\right)}{\left(\sqrt{x+1}+\sqrt{x+2}\right)\left(\sqrt{x+1}-\sqrt{x+2}\right)}dx$

$=\int \frac{\left(\sqrt{x+1}-\sqrt{x+2}\right)}{x+1-x-2}dx$

$=-\int \left(\sqrt{x+1}-\sqrt{x+2}\right)dx$

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

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