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\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}