This paper introduces a new method for numerical integration of a "well-behaved" function over a finite range, which involves expanding the integrand in a series of Chebyshev polynomials and integrating each term separately. The method is compared with commonly used alternatives, such as Simpson's rule and the Gauss method. The advantages of the Chebyshev polynomial method include easy accuracy checking, flexibility in doubling the number of ordinates without additional work, and the ability to provide values of the indefinite integral over the entire range. The paper also discusses the limitations of Simpson's rule and Gauss' formula, particularly in handling infinite ranges and integrals with weak singularities. Special methods like Laguerre-Gauss and Hermite-Gauss are recommended for certain types of integrals over infinite ranges. The introduction of Simpson's rule and Gauss' formula is detailed, including their formulas and practical considerations for implementation.This paper introduces a new method for numerical integration of a "well-behaved" function over a finite range, which involves expanding the integrand in a series of Chebyshev polynomials and integrating each term separately. The method is compared with commonly used alternatives, such as Simpson's rule and the Gauss method. The advantages of the Chebyshev polynomial method include easy accuracy checking, flexibility in doubling the number of ordinates without additional work, and the ability to provide values of the indefinite integral over the entire range. The paper also discusses the limitations of Simpson's rule and Gauss' formula, particularly in handling infinite ranges and integrals with weak singularities. Special methods like Laguerre-Gauss and Hermite-Gauss are recommended for certain types of integrals over infinite ranges. The introduction of Simpson's rule and Gauss' formula is detailed, including their formulas and practical considerations for implementation.