Explore the fascinating world of Gaussian integrals through a comprehensive 3.5-hour tutorial covering twelve different methods of solving this fundamental mathematical concept. Delve into various approaches including polar coordinates, u-substitution, differentiation, Feynman's technique, volume calculations, and the Gamma function. Discover unique perspectives such as the Wallis, Stirling, and Ramanujan methods, as well as applications in complex analysis and Fourier transformations. Investigate challenging variations like the "impossible integral" and "Gauss cubed," and learn to identify common mistakes in Gaussian integral calculations. Master this essential mathematical tool through diverse problem-solving techniques and gain a deeper understanding of its wide-ranging applications in mathematics and physics.