Integral (1)

Let's do some integrals! The good thing about integral is

(a) It requires some tricks to get it done, an interesting intellectual challenge.
(b) It is easy to check if I have got the right answer, even without the model answer, just differentiate it.

Problem:

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

Solution:

 The trick is rationalizing:

$ \begin{eqnarray*} & & \int{\frac{dx}{\sqrt{x} + \sqrt{x + 1}}} \\ &=& \int{\frac{dx}{\sqrt{x} + \sqrt{x + 1}}\frac{\sqrt{x + 1} - \sqrt{x}}{\sqrt{x + 1} - \sqrt{x}}} \\ &=& \int{(\sqrt{x + 1} - \sqrt{x})dx} \\ &=& \frac{2}{3}(x+1)^{\frac{3}{2}} - \frac{2}{3}x^{\frac{3}{2}} + C \end{eqnarray*} $