\begin{frame}
  \frametitle{Antiderivatives / Integrals}

  \begin{exampleblock}{}
    Find the general antiderivatives of 
    \begin{talign}
      f(x) = \frac{1}{x}
    \end{talign}\vspace{-3ex}
    \pause
    
    We have\pause\vspace{-1ex}
    \begin{talign}
      \frac{d}{dx} \ln x = \frac{1}{x}
    \end{talign}
    \pause
    Thus $\ln x + C$
    is the general antiderivative on \pause $(0,\infty)$.
    \pause\medskip
    
    We moreover know that:
    \begin{talign}
      \frac{d}{dx} \ln |x| = \frac{1}{x}
    \end{talign}
    \pause
    Thus $\ln |x| + C$
    is the general antiderivative on intervals not containing $0$.
    \pause
    In particular on intervals $(-\infty,0)$ and $(0,\infty$).
    \pause\medskip
    
    So the general antiderivative of $f$ is:
    \begin{talign}
      F(x) = \begin{cases}
        \ln x + C_1 &\text{for $x > 0$}\\
        \ln (-x) + C_2 &\text{for $x < 0$}
      \end{cases}
    \end{talign}
  \end{exampleblock}
\end{frame}