$\begin{array}{l}
\tan 5x - \tan x = 0\\
\Leftrightarrow \tan 5x = \tan x\\
\Leftrightarrow 5x = x + k\pi \\
\Leftrightarrow x = \dfrac{{k\pi }}{4}\left( {k \in \mathbb{Z}} \right)\\
x \in \left[ {0;\pi } \right) \Rightarrow 0 \le \dfrac{{k\pi }}{4} < \pi \\
\Leftrightarrow 0 \le \dfrac{k}{4} < 1 \Leftrightarrow 0 \le k < 4\\
\Rightarrow k \in \left\{ {0;1;2;3} \right\}\\
\Rightarrow S = 0 + \dfrac{\pi }{4} + \dfrac{\pi }{2} + \dfrac{{3\pi }}{4} = \dfrac{{3\pi }}{2}
\end{array}$
