\begin{frame} \frametitle{The Substitution Rule for Indefinite Integrals} \begin{block}{Substitution Rule} If $u = g(x)$ is differentiable function whose range is an interval $I$ and $f$ is continuous on $I$, then \begin{talign} \int f(g(x)) \alert<2->{g'(x) \,dx} \;=\; \int f(u) \,\alert<2->{du} \end{talign} \end{block} \pause \begin{exampleblock}{} To remember this rule: note that if $u = g(x)$, then \begin{talign} \alert{du = g'(x) dx} \end{talign} (here we think of $dx$ and $du$ as differentials) \end{exampleblock} \pause\medskip In other words: \begin{talign} dx = \frac{du}{g'(x)} \end{talign} \alert{If we change the variable from $x$ to $u = g(x)$ we divide by $g'(x)$!} \vspace{10cm} \end{frame}