Linebreak between \left and \right

if you want the parentheses of the same height, use \vphantom and a combination of \left( ... \right. \\ \left. ... \right)

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\begin{equation}
    \begin{split}
    \left( \int_1^2 ba ba \mathrm da \right. \\
    \left. ba ba ba \vphantom{\int_1^2} \right)% use height of \int
    \end{split}
\end{equation}

\end{document}

Try this:

\documentclass{minimal}
\usepackage{amsmath}
\begin{document}

\begin{equation}
    \begin{split}
    \left( ba ba ba \right. \\
    \left. ba ba ba \right)
    \end{split}
\end{equation}

\end{document}

You were right about the fact that the \left and \right should be on the same line.