Cho hàm số \(y = f\left( x \right)\) thỏa mãn \(f\left( 0 \right) = \frac{2}{3}\) và \(\left( {\sqrt x + \sqrt {x + 1} } \right)f'\left( x \right) = 1\,,\,\,\forall x \ge - 1.\) Biết rằng \(\int\limits_0^1 {f\left( x \right)dx} = \frac{{a\sqrt 2 + b}}\) với \(a,b \in \mathbb{Z}\). Tính \(T = a + b\).

Ta có \(\left( {\sqrt x + \sqrt {x + 1} } \right)f'\left( x \right) = 1{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall x \ge - 1\)

\( \Leftrightarrow f'\left( x \right) = \frac{1}{{\sqrt x + \sqrt {x + 1} }}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall x \ge - 1\)\( \Leftrightarrow f'\left( x \right) = \sqrt {x + 1} - \sqrt x {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \forall x \ge - 1\)

\( \Rightarrow f\left( x \right) = \int {\left( {\sqrt {x + 1} - \sqrt x } \right)dx} = \frac{2}{3}\left[ {\left( {x + 1} \right)\sqrt {x + 1} - x\sqrt x } \right] + C\)

Mà \(f\left( 0 \right) = \frac{2}{3}\) \( \Rightarrow \frac{2}{3}\left( {1 - 0} \right) + C = \frac{2}{3} \Rightarrow C = 0\)\( \Rightarrow f\left( x \right) = \frac{2}{3}\left( {\left( {x + 1} \right)\sqrt {x + 1} - x\sqrt x } \right)\)

Khi đó ta có: \(\int\limits_0^1 {f\left( x \right)dx} = \frac{2}{3}\int\limits_0^1 {\left( {\left( {x + 1} \right)\sqrt {x + 1} - x\sqrt x } \right)dx} \)

\( = \frac{2}{3}.\frac{2}{5}\left. {\left( {{{\left( {x + 1} \right)}^2}\sqrt {x + 1} - {x^2}\sqrt x } \right)} \right|_0^1\)\( = \frac{4}\left[ {\left( {4\sqrt 2 - 1} \right) - \left( {1 - 0} \right)} \right]\)\( = \frac{{16\sqrt 2 - 8}}\)\( \Rightarrow \left\{ {\begin{array}{*{20}{l}}{a = 16}\\{b = - 8}\end{array}} \right.\).

Vậy \(T = a + b = 16 + \left( { - 8} \right) = 8\). Chọn D.