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The integral $\int_{-\infty}^{\infty} e^{- x^2 - g x^4} dx $ is used as anintroductory learning tool in the study of Quantum Field Theory and pathintegrals. Typically it is analysed via perturbation theory. Close formsolutions have been quoted but it is not clear how they were derived. So I setabout deriving the close form solution on my own and using the same methodologyobtain closed form expressions for the even positive integer moments.

Closed-Form Solutions of Zero Dimensional $\phi^4$-Field Theory Using Bessel Functions: A Non-Perturbative Approach