\begin{frame}
  \frametitle{The Substitution Rule for Definite Integrals}

  \subruled
  
  \begin{exampleblock}{}
    \vspace{-1ex}
    \begin{talign}
      \int_1^e \frac{\ln x}{x}\,dx 
    \end{talign}
    \pause
    We choose $u = \pause \ln x$. \pause Then $u' = \pause \frac{1}{x}$\pause, and hence
    \begin{talign}
      \int_1^e \frac{\ln x}{x}\,dx 
      \mpause[1]{&= \int_{u(1)}^{u(e)} \frac{u}{x}\, \frac{du}{1/x} }
      \mpause{= \int_{0}^{1} u\, du }\\
      \mpause{&= \left(\frac{1}{2}u^2\right) \Big]_{0}^{1}}
      \mpause{=  \frac{1}{2}1^2 - \frac{1}{2}0^2}
      \mpause{= \frac{1}{2}}
    \end{talign}
  \end{exampleblock}
  \vspace{10cm}
\end{frame}