Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximations of the integral of a class of sinusoidal composite functions

Alberto Costa

TL;DR

This work tackles the challenge of integrating sinusoidal composite functions, focusing on $C(x)=\int \cos(\cos(x))\,dx$ for which explicit antiderivatives are unavailable. The authors develop two approximations by applying a composite-function integration identity and decomposing the integral into a periodic part and a linear part, first refining the linear term with $J_0(1)$ and then introducing a constant (and later a position-dependent) multiplier for the periodic part. The first method includes a provable error bound, while the second method (with $k(x)=\frac{k_a-k_b}{2}\cos(2x)+\frac{k_a+k_b}{2}$) achieves markedly higher accuracy. Numerical experiments demonstrate that the second approximation yields errors two orders of magnitude smaller than the first and that the evaluation is effectively real-time, making these tools valuable for symbolic computation and scenarios requiring rapid definite integrals of sinusoidal composites. The approach also points to extensions to other forms such as $A\cos(\cos(mx+q))$, broadening applicability in physics and engineering contexts.

Abstract

Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the width of the integration interval. Using such approximations, definite integrals can be computed in almost real-time.

Approximations of the integral of a class of sinusoidal composite functions

TL;DR

This work tackles the challenge of integrating sinusoidal composite functions, focusing on for which explicit antiderivatives are unavailable. The authors develop two approximations by applying a composite-function integration identity and decomposing the integral into a periodic part and a linear part, first refining the linear term with and then introducing a constant (and later a position-dependent) multiplier for the periodic part. The first method includes a provable error bound, while the second method (with ) achieves markedly higher accuracy. Numerical experiments demonstrate that the second approximation yields errors two orders of magnitude smaller than the first and that the evaluation is effectively real-time, making these tools valuable for symbolic computation and scenarios requiring rapid definite integrals of sinusoidal composites. The approach also points to extensions to other forms such as , broadening applicability in physics and engineering contexts.

Abstract

Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the width of the integration interval. Using such approximations, definite integrals can be computed in almost real-time.
Paper Structure (8 sections, 2 theorems, 32 equations, 4 figures, 1 table)

This paper contains 8 sections, 2 theorems, 32 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The integration of $\cos(\cos(x))$
  3. Linear function correction
  4. Periodic function correction
  5. Error analysis
  6. Improved approximation
  7. Numerical experiments
  8. Conclusions and future work

Key Result

Proposition 1

Let $f(x)$ be an integrable function, and $g(x)$ a function for which we can compute both the derivative and the second order derivative. The following equations is valid:

Figures (4)

  • Figure 1: Plots of $\sin(\cos(x))$ (in red - dark) and $\cos(x)$ (in green - light).
  • Figure 2: Plot of the integral of $\cos(\cos(x))$ (in red - dark) and $\tilde{C}(x)$ (in green - light).
  • Figure 3: Plot of $J_0(1)x$ (in red - dark) and the integral of $\cos(\cos(x))$ (in green - light).
  • Figure 4: Plot of $\cos(\cos(x))$ (in red - dark) and $\tilde{P}'(x)+L'(x)$ (in green - light).

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof