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\[I = \int\limits_{ - 1}^0 {\frac{{dx}}{{\sqrt {x + 1} + \sqrt {x + 2} }}} = \int\limits_{ - 1}^0 {(\sqrt {x + 2} - \sqrt {x + 1} )dx} = \left[ {\frac{2}{3}{{(x + 2)}^{\frac{3}{2}}} - \frac{2}{3}{{(x + 1)}^{\frac{3}{2}}}} \right]\begin{cases}0 \\ -1 \end{cases} = \frac{4}{3}(\sqrt 2 - 1)\]
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