36/236
\begin{frame}
  \frametitle{The Substitution Rule for Indefinite Integrals}

  \subrule 
  
  \begin{exampleblock}{}
    \begin{talign}
      \int x^3\cos(x^4 + 2) dx 
    \end{talign}
    \pause
    We choose $u = \pause x^4 + 2$. \pause Then $u' = \pause 4x^3$\pause, and hence
    \begin{talign}
      \int x^3\cos(x^4 + 2) dx 
      \mpause[1]{&= \int x^3\cos(u)\, \frac{du}{4x^3} }
      \mpause{= \frac{1}{4} \int \cos(u)\, du }\\
      \mpause{&= \frac{1}{4} \sin u + C}
      \mpause{= \frac{1}{4} \sin(x^4 + 2) + C}
    \end{talign}
  \end{exampleblock}
  \pause\pause\pause\pause\pause
  \alert{Finding the right $u$ is a guessing game. Often multiple choices.}
  \vspace{10cm}
\end{frame}