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Related Concept Videos

Integration by Parts: Indefinite Integrals01:26

Integration by Parts: Indefinite Integrals

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Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
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Definite integrals involving the product of two functions over a fixed interval can be evaluated using integration by parts. This method rewrites the integral as the difference of a product evaluated at the endpoints and a remaining definite integral that is often simpler to compute.A representative example is the definite integral of the inverse tangent function. Since there is no direct integration formula for arctan ⁡x, the integrand is rewritten as a product of arctan⁡ x and the...
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Improper Integrals: Infinite Intervals01:29

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An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
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Integrals involving powers of tangent and secant are commonly evaluated using substitution, with the strategy determined by the parity of the exponents. The method relies on pairing part of the integrand with the derivative of a suitable trigonometric function and rewriting the remaining factors using trigonometric identities.When the power of secant is even, tangent is chosen as the substitution variable. Since the derivative of tangent is secant squared, a factor of sec⁡2x can be...
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The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing...
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Evaluating Areas Under Curves with DiscontinuitiesA definite integral is considered improper when the integrand is discontinuous at one of the limits of integration. This occurs when the function is undefined or becomes infinite at an endpoint, making the corresponding region under the curve unbounded. Such behavior is commonly associated with vertical asymptotes at the boundary of the interval. To properly define and evaluate these integrals, a limiting process is used to determine whether a...
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Computation of Fresnel Integrals. II.

K D Mielenz1

  • 1Oakland, MD 21550.

Journal of Research of the National Institute of Standards and Technology
|August 24, 2016
PubMed
Summary
This summary is machine-generated.

This study introduces a novel computational method for Fresnel integrals, achieving high accuracy below 1 × 10(-9). The technique enhances numerical analysis by adapting a Boersma approximation for precise calculations.

Keywords:
Fresnel integralscomputationrational approximationsseries expansions

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Area of Science:

  • Numerical Analysis
  • Computational Physics
  • Optics

Background:

  • Fresnel integrals are crucial in optics and wave propagation.
  • Existing methods for computing Fresnel integrals can suffer from accuracy limitations.
  • Boersma's approximate formula offers a basis for improved integral calculations.

Purpose of the Study:

  • To develop an advanced computational method for Fresnel integrals.
  • To achieve a high degree of accuracy, with errors less than 1 × 10(-9).
  • To leverage and adapt existing mathematical approximations for enhanced precision.

Main Methods:

  • The study employs an improved computational approach.
  • The method is derived from a known approximate formula for a related integral.
  • Boersma's formula, as referenced by Abramowitz and Stegun, serves as the foundation.

Main Results:

  • The proposed method significantly enhances the accuracy of Fresnel integral computations.
  • The achieved error rate is demonstrably below the threshold of 1 × 10(-9).
  • The adaptation of Boersma's formula proves effective for precise numerical evaluation.

Conclusions:

  • The developed method offers a highly accurate solution for computing Fresnel integrals.
  • This advancement has implications for fields requiring precise wave propagation analysis.
  • The study validates the utility of adapting established approximations for novel computational challenges.