The Gaussian integral is a fundamental concept in mathematics and physics that represents the integral of the Gaussian function, which is of the form $$I = rac{1}{eta} \\int_{-\\infty}^{\\infty} e^{-\\frac{x^2}{2\\beta^2}} dx$$. This integral is crucial in functional integrals, especially in scalar field theory, as it allows for the evaluation of integrals involving quadratic forms, paving the way for analyzing quantum fields and fluctuations around classical paths.
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