This paper presents a comprehensive study of scalar one-loop integrals in quantum field theory, including one-, two-, three-, and four-point functions, as well as an integral related to soft bremsstrahlung. The authors derive general formulas for these integrals, expressed in terms of Spence functions, which are essential for calculating one-loop radiative corrections in various cases, including those involving unstable particles and particles with spin. The paper begins with an introduction to the challenges of gauge theories and the importance of one-loop calculations in quantum field theory. It then outlines the use of Feynman parameters and propagator identities to simplify the evaluation of these integrals. The authors emphasize the importance of maintaining the correct analytic structure of the amplitudes, particularly in handling branch points and cuts in momentum space. In the one-point function section, the authors derive the general expression for the one-point function, which is valid for arbitrary masses and includes a dimensional regularization method. The two-point function is analyzed using Feynman parameters and involves the evaluation of logarithmic integrals, with careful attention to the analytic structure of the results. The three-point function is derived using a combination of Feynman parameters and propagator identities, leading to an expression involving Spence functions. The authors discuss the validity of the results in different regions of momentum space, including cases with complex masses and unstable particles. The four-point function is the most complex of the integrals discussed, requiring a detailed analysis of the integrand and the use of various transformations to simplify the expression. The authors derive an expression for the four-point function in terms of Spence functions, which is valid for a wide range of momenta and masses, including complex masses with negative imaginary parts. The paper concludes with a discussion of the analytic continuation of the results to different regions of momentum space and the importance of maintaining the correct analytic structure of the amplitudes. The authors emphasize the role of Spence functions in the evaluation of these integrals and the need for careful handling of branch points and cuts in the complex plane. The results presented in this paper provide a solid foundation for the calculation of one-loop radiative corrections in quantum field theory.This paper presents a comprehensive study of scalar one-loop integrals in quantum field theory, including one-, two-, three-, and four-point functions, as well as an integral related to soft bremsstrahlung. The authors derive general formulas for these integrals, expressed in terms of Spence functions, which are essential for calculating one-loop radiative corrections in various cases, including those involving unstable particles and particles with spin. The paper begins with an introduction to the challenges of gauge theories and the importance of one-loop calculations in quantum field theory. It then outlines the use of Feynman parameters and propagator identities to simplify the evaluation of these integrals. The authors emphasize the importance of maintaining the correct analytic structure of the amplitudes, particularly in handling branch points and cuts in momentum space. In the one-point function section, the authors derive the general expression for the one-point function, which is valid for arbitrary masses and includes a dimensional regularization method. The two-point function is analyzed using Feynman parameters and involves the evaluation of logarithmic integrals, with careful attention to the analytic structure of the results. The three-point function is derived using a combination of Feynman parameters and propagator identities, leading to an expression involving Spence functions. The authors discuss the validity of the results in different regions of momentum space, including cases with complex masses and unstable particles. The four-point function is the most complex of the integrals discussed, requiring a detailed analysis of the integrand and the use of various transformations to simplify the expression. The authors derive an expression for the four-point function in terms of Spence functions, which is valid for a wide range of momenta and masses, including complex masses with negative imaginary parts. The paper concludes with a discussion of the analytic continuation of the results to different regions of momentum space and the importance of maintaining the correct analytic structure of the amplitudes. The authors emphasize the role of Spence functions in the evaluation of these integrals and the need for careful handling of branch points and cuts in the complex plane. The results presented in this paper provide a solid foundation for the calculation of one-loop radiative corrections in quantum field theory.