The Jacobi Integral is a mathematical device useful for analysis and solution of some three-body problems. (Three-body problems, like other N-Body Problems, have no general solution.) In cases where it can be used, it produces a constant, i.e., a conserved quantity, sometimes called its Jacobi Constant.
The three-body problems for which it can be used are Restricted Circular Three Body Problems, problems in which two of the bodies are in a circular orbit and the third body has a mass too small to affect them significantly, and Jacobi integral produces an approximation that depends upon this insignificance. Short-term insignificance may not hold over the long term, e.g., Secular motion, so the Jacobi integral has more use regarding short-term situations.