\begin{frame}
  \frametitle{Symmetry}

  Symmetry can sometimes help to simplify integrals!
  \pause
  \begin{block}{}
    Suppose $f$ is continuous on $[-a,a]$:
    \pause
    \begin{itemize}
      \item If $f$ is even [$f(-x) = f(x)$], then
        \begin{talign}
          \int_{-a}^a f(x)\,dx = \mpause[1]{2\int_0^a f(x)\,dx}
        \end{talign}
        \pause\pause\vspace{-1ex}
      \item If $f$ is odd [$f(-x) = -f(x)$], then
        \begin{talign}
          \int_{-a}^a f(x)\,dx = \mpause[1]{0}
        \end{talign}
    \end{itemize}
  \end{block}
  \pause\pause
  
  \begin{exampleblock}{}
    \vspace{-1ex}
    \begin{talign}
      \int_{-1}^1 f(x)\, dx \mpause[6]{\alert{= 0}} &&\text{where}&& f(x) = \frac{\tan x}{1+x^2 + x^4}
    \end{talign}
    \pause
    The function $f$ is \mpause[4]{\alert{odd} since}
    \begin{talign}
      f(-x) \mpause[1]{ = \frac{\sin (-x) / \cos (-x)}{1+(-x)^2 + (-x)^4} }
            \mpause{ = \frac{-\sin x / \cos x}{1+x^2 + x^4} }
            \mpause{ = -f(x) }
    \end{talign}
  \end{exampleblock}
\end{frame}