A new integral formula, which has not been compiled in any integral tables or mathematical softwares, is proposed to obtain the analytical energy-angular spectra of the particles that are sequentially emitted from the discrete energy levels of the residual nuclei in the statistical theory of light nucleus reaction (STLN). In the cases of the neutron induced light nucleus reactions, the demonstration of the kinetic energy conservation in the sequential emission processes becomes straightforward thanks to this new integral formula and it is also helpful to largely reduce the volume of file-6 in nuclear reaction databases. Furthermore, taking p+$^9$Be reaction at 18 MeV as an example, this integral formula is extended to calculate the energy-angular spectra of the sequentially emitted neutrons for proton induced light nucleus reactions in the frame of STLN.
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