Tổng các nghiệm thuộc đoạn

\[2\sqrt 3 cos2\frac{{5x}}{2} + sin5x = 1 + \sqrt 3 \; \Leftrightarrow \sqrt 3 (1 + cos5x) + sin5x = 1 + \sqrt 3 \]

\[ \Leftrightarrow sin5x + \sqrt 3 cos5x = 1 \Leftrightarrow \frac{1}{2}sin5x + \frac{{\sqrt 3 }}{2}cos5x = \frac{1}{2}\]

\[ \Leftrightarrow sin5xcos\frac{\pi }{3} + cos5xsin\frac{\pi }{3} = \frac{1}{2} \Leftrightarrow sin(5x + \frac{\pi }{3}) = sin\frac{\pi }{6}\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{5x + \frac{\pi }{3} = \frac{\pi }{6} + k2\pi }\\{5x + \frac{\pi }{3} = \frac{{5\pi }}{6} + k2\pi }\end{array}} \right. \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = - \frac{\pi }{{30}} + \frac{{k2\pi }}{5}}\\{x = \frac{\pi }{{10}} + \frac{{k2\pi }}{5}}\end{array}} \right.(k \in \mathbb{Z})\)

Với họ nghiệm \[x = - \frac{\pi }{{30}} + \frac{{k2\pi }}{5}\,\,\left( {k \in \mathbb{Z}} \right)\]ta được

\[0 \le - \frac{\pi }{{30}} + \frac{{k2\pi }}{5} \le \frac{\pi }{2} \Leftrightarrow 0 \le - \frac{1}{{30}} + \frac{{2k}}{5} \le \frac{1}{2}\]

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{\frac{1}{{12}} \le k \le \frac{4}{3}}\\{k \in \mathbb{Z}}\end{array}} \right. \Rightarrow k = 1\)

\[ \Rightarrow x = - \frac{\pi }{{30}} + \frac{{2\pi }}{5} = \frac{{11\pi }}{{30}}\]

Với họ nghiệm \[x = \frac{\pi }{{10}} + \frac{{k2\pi }}{5}\,\,\left( {k \in \mathbb{Z}} \right)\]ta được:

\[0 \le \frac{\pi }{{10}} + \frac{{k2\pi }}{5} \le \frac{\pi }{2} \Leftrightarrow 0 \le \frac{1}{{10}} + \frac{{2k}}{5} \le \frac{1}{2}\]

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{ - \frac{1}{4} \le k \le 1}\\{k \in \mathbb{Z}}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{k = 0}\\{k = 1}\end{array}} \right.\)

\( \Rightarrow \left\{ {\begin{array}{*{20}{c}}{x = \frac{\pi }{{10}}}\\{x = \frac{\pi }{{10}} + \frac{{2\pi }}{5} = \frac{\pi }{2}}\end{array}} \right.\)

Vậy tổng các nghiệm thuộc đoạn\[\left[ {0;\frac{\pi }{2}} \right]\]là:\[\frac{{11\pi }}{{30}} + \frac{\pi }{{10}} + \frac{\pi }{2} = \frac{{29\pi }}{{30}}\]