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Fundamentals of integration
Published in Alan Jeffrey, Mathematics, 2004
Hence the constant C is called the constant of integration, and the operation of finding an indefinite integral is called integration. In this notation, the fact that an antiderivative is a function related to the operation of integration, and not just a number as is an ordinary definite integral, is indicated by again employing the integral sign, but this time without limits. On occasions an antiderivative is signified by the notation ∫xf(x)dx, rather than the notation used in Eqn (7.19).
Differential Equations
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
During differentiation, the value of whatever vertical shift is present is lost as a result of the elimination of the constant term, which has a derivative of 0. We normally acknowledge this when integrating a function by adding a +C $ + C $ constant, the constant of integration, to an indefinite integral. This is sometimes a nonissue since, if finding the value of a definite integral, the constant terms cancel and the constant of integration is unnecessary.
Signal Processing Aspects in Wave Propagation
Published in Srinivasan Gopalakrishnan, Elastic Wave Propagation in Structures and Materials, 2023
In many wave experiments, strains are measured instead of displacements and the displacements are extracted by integrating strains. In doing so, if the constant of integration is not considered, then the results obtained will be erroneous. To demonstrate the effect of improper integration of signals, we will consider an infinite rod and will plot the displacement response, as shown in Fig. 6.11 for a triangular input shown in Fig. 4.3.
Solitary wave solutions of nonlinear PDEs using Kudryashov's R function method
Published in Journal of Modern Optics, 2020
Jayita Dan, Sharmistha Sain, A. Ghose-Choudhury, Sudip Garai
Introducing the variable by setting where z = x−vt we obtain after one integration the following ODE: Note that we have set the arbitrary constant of integration to zero. The pole order of (50) turns out to be N = 4 as a consequence of which we assume the solution of (50) to be The substitution of this solution into (50) results in a polynomial of degree eight in R and on equating the coefficients of the different powers of R it turns out that . The remaining equations are given by (Figure 4) The above set of equations admit the following sets of solutions: