  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).
