Definite Integrals Research Articles

Approaches to solving several definite integrals with special functions

Calculus is the foundation of many natural sciences such as Physics. Calculus excels at calculating the area of irregularly shaped objects and thus it may be used in a vast array of domains. Because calculus is difficult to perform when combined with trigonometric and logarithmic functions, additional formulas are required to assist with calculations. By definition, limit of the sum of a function f(x) across given interval [a,b] is the definite integral. Notice the relationship between definite and indefinite integrals is as follow: result of a definite integral is a precise nice value, whereas an indefinite integral is expressed by a function. Their mathematical relationship is limited to computation regarding the Newton-Leibniz formula. This article describes only one of several methods for calculating definite integrals. Taylor expansion will also be used for auxiliary operations, while the relevant equations of Taylor expansion will also be presented in the text. It will also be learned through this paper that the result of the integral varies with .

Lobachevsky's formula and its applications to several definite integrals

The definite integral is a fundamental concept in calculus that has many applications in various fields such as physics, engineering, and economics. However, integration can be difficult and requires a variety of skills such as substitutions and partial integration. In this paper, Lobachevskys formula is explored, which provides a new way to evaluate definite integrals. It should be noted that Lobachevskys formula can only be applied in specific cases where the integrand is even and -periodic. However, it is demonstrated to be an effective method in these cases. In this paper, the proof of the theorem is given, and a variety of examples are solved by virtue of this method. Hence, this paper may serve as a reference for relevant research in the field of calculus and provide insights into the applications of Lobachevskys formula.

Evaluation of limits including integrals by L Hpitals rule

Limit is significant concept in mathematic analysis. Technically, limits definition in mathematics is that a variable in a function gradually approximates to a certain value in the changing process which cannot be ended. L Hpitals rule and Taylor expansion, together with other methods such as Stolz theorem, are usually used in measuring a limits value. In this paper, it will focus on some representative limits that are related to definite integrals. L Hpitals rule and Taylor's expansion are also jointed used so as to solve the problems. The main part of this work talks about the limit of the integration of trigonometric function, under which situation Taylors expansion is commonly utilized. This article talks about the polynomials integration as well, under which situation the approximation method is also employed. Trigonometric function and polynomial function are frequently appeared in evaluating limit. This means that this paper is summarizing the prime functions in integration-related limits.

A brief examination of the atmosphere of exoplanets

As the discovery of exoplanets increased, their characteristics are becoming a main concern for new research. Amongst the many that are discovered, some have an atmosphere of their own just like planet Earth. The purpose of this paper is to examine these atmospheres and calculate the height of the atmosphere using escape velocity. Later in this paper, the issue of the total angular momentum of the planet is examined. This momentum is separated into two different sections whose momentum is added together. The angular momentum of the solid part of the planet is calculated by considering the planet as a rigid body and applying the formula for angular momentum. The second part of calculating the angular momentum is done by using integration to determine the momentum of the atmosphere and using the height of the atmosphere to act as a limit to the definite integral. By understanding the quantities of these exoplanets, further research can be done on how these exoplanets are formed.

Coach as youth development specialist: developing a TPSR-based coach training program and examining participants’ experiences

ABSTRACT Background Effective youth sport coaching encompasses the promotion of personal, social, and athletic skills in a deliberate and consistent manner. Coach training protocols that highlight coaches’ best-practices that promote development emphasize the role of the coach as either a developer of athletes’ personal and social potential or as a developer of athletic talent (focusing on techniques and tactics of sport). However, there is a scarcity of literature describing how to train practitioners for this combined role. Purpose The purpose of this study was to develop and examine coaches’ experiences in coach training that teaches personal and social development, as well as fundamental movement skills called Coach as a Youth Development Specialist (CAYDS). This was conducted in two phases. First, a pilot study aimed to develop, deliver, and evaluate a coach training curriculum that addressed the needs of youth coaches to improve and finalize the CAYDS curriculum. Second, the final iteration of CAYDS was implemented to examine coaches’ lived experiences while participating in the training. Methods The pilot phase involved five-module training delivered to five graduate students. Thematic analysis was used to analyze a focus group interview and understand what participants learned and how the training helped. The results led to a final version of the CAYDS training. In its final version, the training had seven modules that were delivered to a new cohort of six participants. Hermeneutic phenomenology guided the understanding of the what and how of participating coaches’ lived experiences in the CAYDS training. Data were collected using journal reflections, a focus group interview, and individual interviews; and thematically analyzed. Findings The training successfully helped coaches perform coaching competencies, skills, and behaviors identified in the CAYDS framework (cf. Hellison, D. 2011. Teaching Personal and Social Responsibility Through Physical Activity. 3rd ed. Champaign: Human Kinetics; CCE, ASOIF & LMU. 2013. International Sport Coaching Framework v1.2. Champaign: Human Kinetics; SHAPE America. 2019. National Standards for Sport Coaches. 3rd ed. https://www.shapeamerica.org/standards/coaching/default.aspx) that foster personal, social, and athletic development for youth. Findings from this study provide evidence in support of andragogical approaches that helped coaches learn interpersonal, intrapersonal, and professional knowledge (Côté, J., and W. Gilbert. 2009. “An Integrative Definition of Coaching Effectiveness and Expertise.” International Journal of Sports Science & Coaching 4 (3): 307–323.). Participants built positive relationships with community stakeholders and activities involving outreach to parents/guardians, attending school events, and coach educators’ use of role-play were perceived to be particularly impactful. This study also found the process of learning to teach youth skills related to personal and social development inspired coaches to be more self-aware and motivated to grow personally and act in ways that are aligned with TPSR values. Finally, coaches perceived to have acquired professional knowledge that contributed to their ability to instruct fundamental movement skills for athletic development. Conclusions The CAYDS framework offers a viable mechanism to support the dual-role youth sport coaches serve as development specialists, while also parting with coach-centered professional models of coaching. Further, there is promise for this type of development-supportive coaching approach to be applied when working with athletes in their late adolescent years situated in high performance contexts.

New definitions of fractional derivatives and integrals for complex analytic functions

In this paper, we introduce a ground-breaking approach to defining fractional calculus for a selected class of analytic functions. Our new definitions, based on a novel and intuitive understanding of fractional derivatives and integrals, offer improved mathematical tractability for a variety of applications, including physics, engineering and finance. Our approach significantly simplifies the complexity of mathematical functions compared to the traditional Riemann-Liouville approach, by using simple functions rather than special functions, while preserving the intrinsic sense of fractional calculus. This article not only presents our proposed definitions but also provides a thorough analysis of their properties and advantages. The conclusion of this paper discusses the potential for future research in the field of fractional calculus.

Integration on minimal -superspace and emergence of space

We investigate the possibilities of integration on the minimal -superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on despite the fact that the part of the minimal -superspace is

HERONIAN MEAN DERIVATIVE-BASED OPEN NEWTON-COTES QUADRATURE RULES

A novel family of open Newton-Cotes (ONC) formulas is devised for evaluating the definite integrals. The new family is developed by using the Heronian mean in the first-order derivatives of the integrand within the interval [a, b]. The devised Heronian mean derivative-based quadrature rules (HRMDONC) achieve two orders of accuracy enhancement over the conventional ONC quadrature rules. These formulas are derived using the idea of degree of precision. Theorems regarding the degree of precision and order of accuracy are also derived along with the local and global error terms. In addition, the computational order of accuracy of each method is computed confirming the theoretical results. Computational cost and absolute error drops are also determined for three different integrals from the literature which demonstrate the superiority of the proposed HRMDONC methods over the classical ONC.

TRIPLE SERIES EVALUATED IN π AND LN 2 AS WELL AS CATALAN’S CONSTANT G

By computing definite integrals, several infinite triple series are explicitly evaluated in terms of @ and ln2 as well as Catalan’s constant G.

A Bi−directional method for evaluating integrals involving higher transcendental functions. HyperRAF: A Julia package for new hyper−radial functions

The electron repulsion integrals over Slater−type orbitals with non−integer principal quantum numbers are investigated. These integrals are useful in both non−relativistic and relativistic calculations of many−electron systems. They involve hyper−geometric functions that are practically difficult to compute. Relationships free from hyper−geometric functions for expectation values of Coulomb potential (r21−1) are derived. These relationships are new and show that the complication coming from two−range nature of Laplace expansion for the Coulomb potential is removed. This is achieved by utilizing auxiliary functions represented in finite power series. They serve as essential components in deriving straightforward recurrence relationships for electron repulsion integrals. In the context of computing the expectation values of potentials with arbitrary power, the methodology presented here for evaluation of these integrals forms the initial condition. It is also adapted to multi−center integrals. Program summaryProgram Title: HyperRAFCPC Library link to program files:https://doi.org/10.17632/6pbv2y7s42.1Developer's repository link:https://github.com/abagciphys/HyperRAF.gitLicensing provisions: MITProgramming language:Julia Programming Language [1]Supplementary material: An exploratory variant of the software program written in the Mathematica Programming Language [2].External routines/libraries:Nemo, a computer algebra package for the Julia programming language [3], JRAF, a Julia package for computation of relativistic molecular auxiliary functions [4].Nature of problem: Definite integrals involving higher transcendental functions given by,fmn1(a,b,x)=xm−1e−bxΓ[n,ax], andfmn2(a,b,x)=xm−1e−bxγ[n,ax] are frequently encountered in atomic physics, with the electron repulsion integral being a notable illustration.For exclusive solutions to these integrals, one can refer to Erdélyi's [5] or Gradshteyn and Ryzhik's [6] books. The solutions are obtained by using the series representation of incomplete gamma functions. The result is hyper−geometric functions of the form F12[1,b,c;z], where b=m+n, c=m+1, z=b/(a+b) for fmn1(a,b,x) and c=n+1, z=a/(a+b) for fmn2(a,b,x), respectively. Due to the non−trivial structure of infinite series that are used to define them, the computation for hyper−geometric poses challenges. Convergence of their series strictly depends on the values of parameters. Computational issues such as cancellation or round−off error emerge.Solution method: This research introduces novel bi−directional hyper−radial functions that are used to establish fresh recurrence relationships for electron repulsion integrals, eliminating the dependence on hyper−geometric functions. Dual functionality is inherent in the hyper−radial functions as they offer alternative solutions for definite integrals involving higher transcendental functions. Additionally, they transform the representation of the hyper−geometric functions into finite power series.

Vulnerable Integrity

This paper presents a social-theologically informed interpretation of the term integrity, as it occurs in fundamental law. It explores the manifestations of integrity violations and proceeds to draw an inference: an integrity violation can directly emanate from a misconception regarding integrity itself, as well as the implementation of protective measures that follow it. Integrity in its wholeness dimension is understood as open-endedness and non-seclusion rather than as a substantial, clearly definable characteristic of a person. This open-endedness and non-seclusion results from the relational constitution of an individual. Consequently, it follows that a violation of integrity occurs when the open-endedness and non-seclusion of a person and their relational Becoming is hindered. The new definition of integrity is particularly important when it comes to the governance of new health technologies, especially Digital Twins that can become representatives of a person. Human integrity is non-violated only when it is understood as open to relational Becoming and this Becoming shows its expression in the mutual enabling and support of self-articulation.

Method for determining eigen numbers in heat conduction problems for a cylinder

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

The Story of Definite Integrals: A Calculus Textbook Narrative Analysis

There is a plethora of textbooks to choose from. While most textbooks contain the same content, teachers need to understand the textbook they have chosen to support their students’ learning, as textbooks can significantly impact student learning and teacher instruction. When searching for a way to understand the various textbooks, my research question became, How do calculus textbooks tell the story of definite integrals? My research is in response to the never-ending search for how teachers can select and use their resources. Also, this study contributes to understanding the relationships between stories and knowledge construction using Dietiker’s (2015) narrative framework to analyze the written calculus curriculum. I wrote the specific stories of introducing the definite integral as told by five calculus textbooks. Analyzing how the stories developed disclosed the similarities and differences among the texts in teaching definite integrals. The analysis also revealed a variety of avenues to introduce and teach definite integrals, including a focus on the area under the curve, Riemann sums, or the Fundamental Theorem of Calculus, and some implications to the context of physics related to these avenues. These insights provide teachers with a deeper understanding of textbook lessons and the variances that can potentially alter student learning.

How do students learn definite integrals? Exploring students' learning opportunities

AbstractThis study explores calculus students' opportunities to learn the concepts of integral by examining one mathematician's videotaped lessons and the textbook. Results show that both lessons and the textbook introduce important cognitive resources briefly and focus on other units of knowledge. Implications to these results are also discussed.

Back from the rabbit hole. Theoretical considerations and practical guidelines on psychedelic integration for mental health specialists.

The growing interest in and prevalence of the use of psychedelics, as well as the potential benefits and negative consequences associated with psychedelic experiences, create a need for mental health specialists to be able to provide adequate and effective intervention regarding the content and consequences of these experiences, that is, psychedelic integration. At the same time, current graduate training in psychiatry, psychology, psychotherapy, counseling, etc., fails to adequately prepare professionals for such interventions. In order to fill this gap, an international, bottom-up project was established to attempt developing guidelines. This project was conducted by means of literature reviews as well as roundtable discussions among project participants, leading to a consensus on the guidelines' final scope and content. Drawing from the outcomes of this project, this article presents proposed comprehensive guidelines covering both theoretical and practical aspects of psychedelic integration, that are intended to serve as a resource for various mental health specialists who may encounter individuals in need of support considering their psychedelic experiences. These guidelines encompass clinician-friendly information on the effects of psychedelics, a definition of psychedelic integration, the general theoretical considerations linked to utilization of psychedelic experiences in clinical practice, a simple model organizing the course of psychedelic integration practice, as well as an overview of the current models of psychedelic integration, along with a selective presentation of basic and specific interventions derived from various psychotherapeutic approaches that can be employed in the practice of psychedelic integration.

107.35 Two definite integrals that are (not surprisingly) equal

107.35 Two definite integrals that are (not surprisingly) equal

Conflict personalization: a systematic literature review and the development of an integrative definition

PurposeScholars and practitioners alike wish to understand what makes workplace conflict beneficial or injurious to, for example, performance and satisfaction. The authors focus on parties’ personal experience of the conflict, which is complementary to studying conflict issues (i.e. task- or relationship-related conflict). Although many authors discuss the personal experience of conflict, which the authors will refer to as conflict personalization, different definitions are used, leading to conceptual vagueness. Therefore, the purpose of this paper is to develop an integrative definition of the concept of conflict personalization.Design/methodology/approachThe authors conducted a systematic literature review to collect definitions and conceptualizations from 41 publications. The subsequent thematic analysis revealed four building blocks that were used to develop an integrative definition of conflict personalization.FindingsThe authors developed the following definition: Conflict personalization is the negative affective as well as cognitive reaction to the self being threatened and/or in danger as a result of a social interaction about perceived incompatibilities.Practical implicationsThe integrative definition of this study enables the development of a measurement instrument to assess personalization during workplace conflict, paving the way for developing effective research-based interventions.Originality/valueConceptual vagueness hampers theoretical development, empirical research and the development of effective interventions. Although the importance of conflict personalization is mentioned within the field of workplace conflict, it has not been empirically studied yet. This paper can serve as the basis for future research in which conflict issue and personal experience are separated.

What is integrative medicine? Establishing the decision criteria for an operational definition of integrative medicine for general practice health services research in Australia

BackgroundOperational definitions outline how a conceptual definition will be measured for consistent, reproducible data collection and analysis. This article reports the decision criteria that will be used for an operational definition of integrative medicine (IM) in a secondary analysis of an Australian national survey of general practitioner activity. MethodsA multidisciplinary team applied an iterative approach, informed by expert knowledge and literature reviews to establish decision criteria for categorizing the terms in the Australian clinical interface terminology of the International Classification of Primary Care, second edition (ICPC-2 PLUS) and the Coding Atlas for Pharmaceutical Substances, according to whether they reflected IM, conventional/mainstream medicine (MM), or both IM and MM (IM/MM). ResultsThe final decision criteria categorized all terms for examinations, investigations, advice/counselling, and drugs with synthetic ingredients, and terms for referrals to secondary care services and healthcare practitioners that are not a traditional or complementary medicine practitioner as MM. Terms that could apply to both styles of clinical practice (e.g., preventive health, lifestyle medicine, psychosocial and some drugs with natural ingredients) were categorised as IM/MM. The remaining terms, that mostly reflected the World Health Organization's theoretical definitions of traditional and complementary medicine, were categorized as IM. ConclusionDifferentiating between integrative and conventional/mainstream medicine in general practice is context specific and not always possible. The category IM/MM proposes integrative medicine as an extension, rather than an alternative. The rationale for the integrative medicine operational definition has relevance for researchers and health services in Australia, and internationally.

Medical marketing in healthcare institutions with public-private partnership mechanisms

The aim – to reveal the essence of medical marketing in healthcare institutions (HCI) with principles of public-private partnership (PPP). The study used general scientific methods of analysis, synthesis, generalization, interpretation of scientific data, as well as systemic and structural-functional approaches. The article reveals the essence, principles and functions of medical marketing in HCI with the mechanisms of PPP. The substantiation of the main marketing tools is given – business processes and integration processes that contribute to effective marketing in HCI. It also identified the key points that form the structure of medical business processes. The process approach to business process management is disclosed, which allows getting benefits for HCI. The definition and significance of the medical processes integration in HCI on PPP principles is provided, which contributes to the coordination of marketing activities with the strategy development within each integrated complex of HCI and the possibility of obtaining a positive synergistic effect. Examples of internal and external integration are given. The system for managing the external integration process is shown. To stimulate the integration process, bundled payment method has been defined.Conclusions. The article provides justification for medical marketing in HCI with the principles of PPP, focused on business processes and the integration of medical services, which ensures economic, medical and social efficiency and optimal functioning of HCI in market conditions.

Epistemic decentering in education for responsibility: revisiting the theory and practice of educational integrity

There is no consensus on definitions of educational or academic integrity, and their philosophical relationship with the notion of responsibility is complex. Here, we aim to i) disentangle these three notions. We lean on a philosophical framework of ethics and our method involves different kinds of reasoning and the modeling of complex thinking. We combine this frame with a three-level epistemic dimension to allow us ii) to model the psycho-epistemic (level 1), epistemological (level 2), and phenomenological (ground 0) ways in which subjects interact with their own norms and knowledge and with those of the surrounding institutions. Finally, iii) we also aim to propose concrete educational means by which to implement educational integrity. Our theoretical findings lead us i) to consider responsibility as a process that consists of establishing a dialogical relationship between one’s inner and outer worlds, which relies on an epistemic decentering. Based on this, we argue that education for responsibility founds a new, expanded definition of educational integrity. Moreover, ii) empirical evidence suggests that this model can be operationalized by psychological indicators such as critical and complex thinking, cognitive flexibility, contextual relativism, and decentering, all of which are skills that can be fostered in spite of simplifying thinking, dogmatism, naive epistemology (and dualism) and cognitive fusion, respectively. It points to iii) the benefits of an educational approach in which subjects are encouraged to practice different types of meditation and to feel free to break institutional rules. Therefore, promoting educational integrity may require methods that lie beyond the obvious choices. After discussing the scope and limitations of our results, we propose a new research agenda for educational integrity, which could ground a field of research broader than just academic integrity, but complementary to it.