  In this paper, we have proposed a new method for solving the Gaussian
integral. Introducing a parameter that depends on a $n$ index, we have found a
general solution for this type of integral inspired by Taylor series of a
simple function. We have demonstrated that this parameter represents the Taylor
series coefficients of this function, a result very newsworthy. We have also
introduced some Theorems that are proved by mathematical induction. The
proposed method in this work has shown more practical and accessible than some
methods found in the literature. As a test for the method, we have investigated
a non-extensive version for the particle number density in Tsallis framework,
which enabled us to evaluate the functionality of the method. Besides,
solutions for a certain class of the gamma and factorial functions are derived.
Moreover, we have presented a simple application in fractional calculus. In
conclusion, we believe in the relevance of this work because it presents a new
form of solving the Gaussian integral having the differential calculus as a
tool.
