
      \textbf{4.} Spočtěte integrál
      $$
            \int \frac{x\,dx}{x^8 - 1}.
      $$

      \textbf{Řešení} Použijeme substituci $x^2 = t$, $2x \, dx = dt$:
      \begin{multline*}
            \int \frac{x\,dx}{x^8 - 1} \subst
            \frac12 \cdot \int \frac{dt}{t^4 - 1} =\\
            =\frac12 \cdot \int \frac{dt}{(t^2-1)(t^2 + 1)} =
            \frac14 \cdot \int \left( \frac 1{t^2-1} - \frac 1{t^2+1} \right) dt\\
      \end{multline*}
      Použijeme vzorec $\int \frac{dx}{1-x^2} \konst \frac12 \log \left\lvert \frac{1+x}{1-x} \right\rvert$:
      \begin{multline*}
            \frac14 \cdot \int \left( \frac 1{t^2-1} - \frac 1{t^2+1} \right) dt \konst
            -\frac14\log\left\lvert\frac{1+t}{1-t}\right\rvert - \frac14\arctg t \subst\\
            \subst -\frac18\log\left\lvert\frac{1+x^2}{1-x^2}\right\rvert - \frac14\arctg x^2.
      \end{multline*}
