    \textbf{3.} Spočtěte integrál
    $$
        \int\frac{1}{x^2+x+1} \dx.
    $$

    \textbf{Řešení}
    \begin{multline*}
        \int\frac{1}{x^2+x+1} \dx =
        \int\frac{1}{ \left( x+\frac12 \right)^2 + \left(\frac{\sqrt3}{2}\right)^2 } \dx =\\
        \int\frac{1}{ \left(\frac{\sqrt 3}{2}\right)^2 \biggl(\left(\frac{2x}{\sqrt3} =
            \frac{1}{\sqrt 3}\right)^2 + 1\biggr) } \dx \subst
        \begin{bmatrix}
            \frac{2x+1}{\sqrt3} = u\\
            \frac{2}{\sqrt3} \dx = \dnu\\
            \dnx = \frac{\sqrt3}{2} \du
        \end{bmatrix}\subst\\
        \subst\frac{2}{\sqrt3} \int\frac{1}{u^2 + 1} \du \konst
        \frac{2}{\sqrt3} \arctg u \subst
        \frac{2}{\sqrt3} \arctg\left(\frac{2x}{\sqrt3} + \frac{1}{\sqrt3}\right).
    \end{multline*}
