Simplifying and extending a useful class of signals and impulse responses

Mathematics is an essential tool for studying science and engineering, and calculus is one of the most important branches of mathematics for engineering. In this paper a new formula for evaluating /spl int/x/sup n/e/sup ax/dx and a more generally applicable extension to polynomials are developed. This new approach illustrates the intimate relationship between differentiation and integration, and is simple enough for a freshman taking the first course in calculus to derive it. Although a closed-form expression for this integral exists, it is cumbersome and relatively more difficult to remember than the forms proposed in the paper. Also, the proposed formula readily generalizes to a larger class of polynomials, thus becoming much more useful. We show that these formulae are particularly important for the analysis and use of a broad class of signals commonly encountered in the classroom and in practical situations. The proposed formulae are applied to Fourier Series, Fourier Transforms, Laplace Transforms, and time domain convolution.