 \textbf{3.} Spočtěte integrál
      $$
            \int \frac{dx}{x(x^{10} + 2)}.
      $$

      \textbf{Řešení} Nejprve upravíme výraz:
      $$
            \int \frac{dx}{x(x^{10} + 2)} = \int \frac{x^9}{x^{10}(x^{10} + 2)} dx.
      $$ Teď použijeme substituci $x^{10} = t$, $10x^9 dx = dt$:
      \begin{multline*}
            \int \frac{dx}{x(x^{10} + 2)} =
            \int \frac{x^9}{x^{10}(x^{10} + 2)} dx \subst\\
            \subst\frac{1}{10} \int \frac{ dt }{ t(t + 2) } =
            \frac{1}{20} \int \left( \frac{1}{t} - \frac{1}{t+2} \right) dt \konst\\
            \konst\frac{1}{20}\log \lvert t \rvert - \frac{1}{20}\log \lvert t+2 \rvert \subst
            \frac{1}{20}\log x^{10} - \frac{1}{20}\log (x^{10} + 2).
      \end{multline*}
