\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}