Question

Integrate \[ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+2}}. \]

Hint:

Rationalize denominators involving sums of square roots to simplify the integral.

Answer 1

Rationalize the denominator: \[ \frac{1}{\sqrt{x+1} + \sqrt{x+2}} = \frac{\sqrt{x+2} - \sqrt{x+1}}{(\sqrt{x+2} + \sqrt{x+1})(\sqrt{x+2} - \sqrt{x+1})} = \frac{\sqrt{x+2} - \sqrt{x+1}}{(x+2) - (x+1)} = \sqrt{x+2} - \sqrt{x+1}. \] So the integral becomes: \[ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+2}} = \int \left(\sqrt{x+2} - \sqrt{x+1}\right) dx = \int \sqrt{x+2} \, dx - \int \sqrt{x+1} \, dx. \] Integrate each term separately: \[ \int \sqrt{x+a} \, dx = \frac{2}{3} (x+a)^{3/2} + C. \] Therefore, \[ \int \sqrt{x+2} \, dx = \frac{2}{3} (x+2)^{3/2} + C_1, \] and \[ \int \sqrt{x+1} \, dx = \frac{2}{3} (x+1)^{3/2} + C_2. \] Hence, \[ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+2}} = \frac{2}{3} (x+2)^{3/2} - \frac{2}{3} (x+1)^{3/2} + C. \]
Final answer: \[ \boxed{ \int \frac{dx}{\sqrt{x+1} + \sqrt{x+2}} = \frac{2}{3} \left[(x+2)^{3/2} - (x+1)^{3/2}\right] + C. } \]