Integration of e^(x^n) and e^(-x^n) in forms of series, their   applications in the field of differential equation; introducing generalized   form of Skewness and Kurtosis; extension of starling's approximation
  In this paper we tried a different approach to work out the integrals of e^(x^n) and e^(-x^n). Integration by parts shows a nice pattern which can be reduced to a form of series. We have shown both the indefinite and definite integrals of the functions mentioned along with some essential properties e.g. conditions of convergence of the series. Further more, we used the integrals in form of series to find out series solution of differential equations of the form x[(d^2 y)/(dx^2)]-(n-1)(dy/dx)-n^2 x^(2n-1)y-nx^n=0 and x[(d^2 y)/(dx^2)] -(n-1)(dy/dx)-n^2x^(2n-1)y+(n-1)=0, using some non standard method. We introduced modified Normal distribution incorporating some properties derived from the above integrals and defined a generalized version of Skewness and Kurtosis. Finally we extended Starling's approximation to limit [n to infinity ] (2n)! ~ 2n * \sqrt{(2\pi)} [(2n/e)]^(2n). 
T. S. Konar and S. Paul