Generated by the Physics Derivation Graph. Eq. \ref{eq:3494854} is an identity. \begin{equation} d(u v) = u dv + v du \label{eq:3494854} \end{equation} Subtract \(v du\) from both sides of Eq. \ref{eq:3494854}; yields Eq. \ref{eq:2938188}. \begin{equation} d(u v) - v du = u dv \label{eq:2938188} \end{equation} Swap LHS of Eq. \ref{eq:2938188} with RHS; yields Eq. \ref{eq:2938190}. \begin{equation} u dv = d(u v) - v du \label{eq:2938190} \end{equation} Indefinite integral of both sides of Eq. \ref{eq:2938190}; yields Eq. \ref{eq:3848329}. \begin{equation} \int u dv = u v - \int v du \label{eq:3848329} \end{equation} Eq. \ref{eq:3848329} is one of the final equations.