IN THIS PAPER, WE STATE AND PROVE A NEW FORMULA EXPRESSING EXPLICITLY THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OF THE TRIANGULAR FUNCTIONS (TFS) WITH ANY FRACTIONAL-ORDER, IN TERMS OF TFS THEMSELVES AND USE IT TO CONSTRUCT A NEW AND GENERAL FORMULATION FOR THE TRIANGULAR FUNCTION (TF) Operational matrix OF FRACTIONAL INTEGRAL (TF-OMFI). ALSO, BY USING THE TF-OMFI AND A SPECTRAL TAU METHOD, WE DEVELOP A DIRECT SOLUTION TECHNIQUE FOR SOLVING A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION.