\begin{frame}
  \frametitle{The Substitution Rule for Indefinite Integrals}
  
  How to find antiderivatives of formulas like
  \begin{talign}
    \int 2x\sqrt{1 + x^2} dx \quad\text{?}
  \end{talign}
  \pause
  
  Recall the chain rule:
  \begin{talign}
    \big( f(g(x)) \big)' \;=\; \alert<3->{f'(g(x)) g'(x)}
  \end{talign}
  \pause
  Let us try to write the integral in the form:
  \begin{talign}
    \int f'(g(x)) g'(x) dx
  \end{talign}
  \pause
  Then 
  \begin{talign}
    f'(x) = \mpause[1]{\sqrt{x}}
    &&
    g(x) = \mpause{1 + x^2}
    &&
    g'(x) = \mpause{2x}
  \end{talign}
  \pause\pause\pause\pause
  Moreover the antiderivative of $f'(x)$ is \quad $f(x) = \pause \frac{2}{3}x^{\frac{3}{2}}$
  \pause. Thus
  \begin{talign}
    \int 2x\sqrt{1 + x^2} dx = \mpause[1]{ f(g(x)) + C } \mpause{= \frac{2}{3}(1+x^2)^{\frac{3}{2}} + C}
  \end{talign} 
  \vspace{10cm}
\end{frame}