Some Aspects of Teaching Improper Integrals in an Electronic Environment

Abstract

The development of computer technology and communication systems provides new opportunities for learning in an electronic environment. Remote education in virtual electronic classrooms has been widely used in recent decades, especially during the pandemic. Teaching in an electronic environment makes it possible to conduct lectures with thousands of participants. This article is devoted to a new look at teaching improper integrals at the undergraduate and postgraduate levels. A new partition of unity method is applied in a deep recursive e-learning process. A thorough list of subproblems of a real model problem is demonstrated. Additionally, other appropriate improper integrals are considered.

1. Introduction

The improper integral was created in the first part of the seventeenth century. Due to the various engineering applications of improper integrals, the development of integral calculus has enjoyed great interest in the last decades.

Broadly speaking, we classify the papers on improper integrals into six different groups:

• papers devoted to the theory of teaching [1,2,3,4];
• papers dealing with new educational technologies including the electronic environment [5,6];
• studies that deal with student mistakes and difficulties [6,7,8,9];
• research on curricula [2,4];
• papers that consider new methods for exact calculations of improper integrals [10,11,12];
• papers introducing new rules for numerical integration [13].

We will next mention some efficient approaches to teaching improper integrals.

Arfi and Wiryanto [14] have analyzed common errors found in solving improper integrals. They emphasized the significance of correctly notating the definition of improper integrals in calculus. Failure to do so can lead to unexpected outcomes in calculations, highlighting the importance of precise notation in this area of mathematics.

Mateus-Nieves and Hernández Montañez [5] have studied the challenges associated with learning improper integrals in university teaching practice and analyzed the results of using mathematical software. Also, they have emphasized the need for further work on understanding the concepts and the significance of the fundamental tools necessary for correctly understanding these integrals.

Bressoud et al. [15] have provided a summary of current research trends in calculus education. They have discussed key issues such as students’ difficulties, classroom practices, task design, and the dialectical relationship between students’ thinking and institutional expectations. The paper highlighted the importance of epistemological considerations in addressing tensions within calculus education and emphasized the importance of teaching practices in overcoming these challenges. Furthermore, it raised questions about the impact of task design on students’ learning outcomes, the influence of teachers’ beliefs on implementation, and the interaction between design and classroom time allocation. The study also suggested that research on digital technology tools in mathematical education could enhance calculus education. Overall, the text offered insights into recent developments in calculus education research and proposed new avenues for future investigation.

Gonzalez-Martin and Correia De Sa [16] have discussed the design and effectiveness of a teaching sequence to improve students’ understanding of improper integrals. It focuses on integrating graphical methods into education, drawing from historical examples, and analyzing the effectiveness of the teaching strategy based on collected data and student feedback. They have considered the topic of improper integrals in three important aspects: didactic, cognitive, and epistemological dimensions.

Galán-García et al. [17] presented the difficulties in computing indefinite integrals in an engineering context. They introduced advanced methods and techniques for solving these problems. The authors have discussed the use of Computer Algebra Systems (CASs) for this purpose. They presented rules and methods for improving the capabilities of CAS to deal with indefinite integrals, and conducted tests to evaluate the performance of various CASs in this area.

Browsing published research up to now, we found a small number of papers on new theoretical approaches for teaching improper integrals. Additionally, there is a critical lack in papers dealing with the theory of teaching improper integrals in the electronic environment. Nowadays, there are various software platforms such as MS Teams, Zoom, and Google Classroom, where every lecturer can easily build an electronic classroom. The education in an electronic environment has the following basic advantages:

• the lecturer can teach a large number of students (even thousands);
• the lecturer is supported by unbounded resources from the web during the teaching process;
• the students can simultaneously cooperate in different groups in the exercise classes;
• the students can create dynamic peer-to-peer relations for solving specific problems.

Therefore, the necessity of new educational technologies for teaching improper integrals in an electronic environment arises naturally.

This paper deals with a new approach to teaching improper integrals in an electronic environment. To this end, a deep recurrent process is organized, creating a new partition of unity method. The new approach is thoroughly demonstrated by an appropriate model problem. Examples of other appropriate model problems are included.

The paper is organized as follows. A new teaching approach based on the partition of unity method is presented in Section 2. The idea is illustrated by a recursive algorithm and two figures. The theory is supported by a thoroughly analyzed example in Section 3. There, the main problem is divided into a list of subproblems. Some concluding remarks are presented in Section 4.

2. Materials and Methods

The main goal of this research is to create an e-learning algorithm where the students are active subjects of the learning process. Nowadays, the teaching should not be restricted to a fixed classroom. Electronic classrooms [18,19] have been successfully used in the last decades, especially during the pandemic. Conducting online courses with practical exercises and feedback is an advanced approach in contemporary education. One of the most important criteria for the quality of the learning process consists of to what extent the students absorb the learning content in the lecture and exercise classes and to what extent they can apply the learned strategies to solve practical problems.

Let us concentrate on the topic of improper integrals. A wide amount of prior knowledge is required for students to solve improper integrals. Often, the improper integrals describe integral operators and linear functionals that are directly related to the solution of various boundary value problems. That is why the control over the learning process is of considerable practical importance. We propose a partition of unity method to control the different stages of the learning process. We distinguish four different stages in teaching students on a fixed thematic unit:

• teaching to obtain new knowledge;
• lectures to establish the relationships between the new notions and their properties;
• students’ activities to apply the new knowledge in solving computational problems;
• exercises to solve mathematical models of real-life problems based on improper integrals.

Our considerations are related to the third stage of the teaching procedure. There are three basic problems related to improper integrals:

• proving the convergence;
• exact computations;
• approximate computations.

We focus our research on methods for teaching the exact calculations of improper integrals. We suppose that the students are not located in a fixed classroom. The students communicate with the lecturer and each other via computer devices and software programs. We denote the object of interest by $P$. The problem $P$ could be related to a linear or a nonlinear integral operator, a linear functional, a sequence of nested improper integrals, a weak fractional Laplacian, or Laplace’s operator, for instance. In some specific cases, we could ask students to directly solve improper integrals of unbounded domains or improper integrals of unbounded functions. In preliminary preparation for the application of the proposed theory, it is necessary to consider a large number of examples of the transformation of improper integrals of a discontinuous integrand into integrals with infinite bounds of integration.

We continue with thoroughly explaining the partition of unity method, which is essentially applied in our teaching process. This method is addressed to well-motivated students with a strong background in the area of interest. The main goal of the method is for students to consolidate their knowledge of a given theoretical unit. The method requires a set of multistage problems necessary for students’ standalone works. Each multistage problem is partitioned into clearly defined indivisible subproblems. The students are directed to consider each subproblem as a separate task. All subproblems are provided with the necessary theoretical background. The students solve the sequence of the subproblems independently and receive theoretical or practical hints only in the case of nonsuccess to a fixed stage. After the students present a solution to all subproblems, the teacher should present all known solutions to each task, making a comparison between individual approaches to its implementation. In this case, comparisons should be made based on the theory used. Finally, students should present all possible decompositions of the main problem into indivisible subproblems. If a student cannot solve the $i$th subproblem, they should not stop solving the main problem. This student should continue with the next subproblem, assuming that the $i$th subproblem has already been solved. To continue with the solution of the $(i+1)$th subproblem, the student receives the answer to the $i$th one from the teacher. This transition is a crucial point when applying the partition of unity method. At the end, the students only carry out various subdivisions of multistage problems and compare the different partition techniques.

The main goal in the third stage of the teaching process is to compile a problem $P$ that needs theoretical units that are as large as possible. To apply our partition of unity method, we divide the main problem $P$ into a large number of easily solvable subproblems $P_i,$ $i=\mathrm{1,2},\dots ,n$. The individual subproblems should be practically indivisible. The lecturer prepares a knowledge base as follows:

$$K_i=\left\{K_{i1},K_{i2},\dots ,K_{i{\nu }_i}\right\}$$

for all $P_i$. Each knowledge base comprises basic concepts, definitions, theorems, properties, rules, tables, basic inequalities, etc., related to the subproblem $P_i$. The elements $K_{ij}$ of the knowledge base $K_i$ satisfy $K_{ij}\subset K_{ij+1}, j=\mathrm{1,2},\dots ,{\nu }_i-1.$ A ranking procedure for the solution to each subproblem should be introduced. On the other hand, the subproblems should be ranked by difficulty. It is important to allocate a specific time $t_i$ for solving each subproblem $P_i$. The time given to solve a subproblem is based on the amount of work required and the level of difficulty of the respective subproblem. Furthermore, it is crucial to consider the relationship between the time required to solve the main problem and the total amount of work involved. The choice of problem $P$ should guarantee that the overall necessary time

$$T\left(P\right)=\sum _{i=1}^n{t}_i\left(P_i\right)$$

(1)

is no more than the time for the planned daily exercise regarding the curriculum. The students should be compared according to the solution of each subproblem. Before sending the written assignment to the students, some initial discussion with respect to the nonlocal properties of integral operators and linear functionals should be conducted.

There are two different options: the students communicate with a lecturer who is a real person, or the education is conducted via computer software located on the server. Both of them have advantages and disadvantages. We assume that the learning process is operated by a supervisor who can be a lecturer or an educational software located on a university server. Initially, the supervisor sends the main problem $P$ to all students. He gives indications of how the problem $P$ can be divided into subproblems and presents a plan for solving the problem.

Since the procedure for all subproblems is the same, we consider the processing of the $i$th subproblem. The supervisor requires the students to submit a solution to the $i$th subproblem $P_i$ within the time $t_i$ regulated by (1). All students have to send the solutions while keeping the deadline. After receiving the solutions, the supervisor ranks the solutions of the students and separates three groups:

• the group $B_i$ comprises all students with a full solution to the subproblem $P_i$;
• the group $W_i$ consists of all students without a solution to the subproblem $P_i$;
• the representatives of group $I_i$ have incomplete solutions to $P_i$.

Every subproblem $P_i$ generates different groups $B_i, W_i$, and $I_i$ of students. The supervisor sends an initial base of knowledge $K_{i1}$to each student $W_{ir}$. The set of knowledge $K_{i1}$ is intended to support the activities of student $W_{ir}$ in solving the subproblem $P_i$. The student $W_{ir}$ establishes peer-to-peer communication with a colleague $B_{is}$ to obtain help in solving $P_i$. We denote this communication by $W_{ir}\rightleftarrows B_{is}$. In the case of no success, the set of knowledge $K_{i1}$ should be gradually expanded to $K_{i{\nu }_i}$. The expansion of $K_{i1}$ is individual for each student and depends on the difficulties they encounter in the task at hand. If student $W_{ir}$ is again unable to prepare a complete solution to subproblem $P_i$, then student $B_{is}$ submits a full solution to $P_i$ to $W_{ir}$. Moreover, in this case, $B_{is}$ explains verbally the solution to $P_i$. The process of education is illustrated by Figure 1 and Figure 2.

We apply a different approach to the students from the group $I_i$. The supervisor creates a different base of knowledge for each student regarding their individual difficulties. All students from group $I_i$ establish client–server communications with the supervisor. If the group of all students is well-distributed, we have the following:

$$\mathrm{Card}\left(W_i\right)\approx \mathrm{Card}\left(B_i\right).$$

(2)

The cardinality of the group $I_i$ is not important, because the students from this group communicate directly with the supervisor. The cases

$$\mathrm{Card}\left(W_i\right)\ll \mathrm{Card}\left(B_i\right)$$

and (2) are trivial and easy to implement. Let us consider a badly distributed group of students with

$$\mathrm{Card}\left(W_i\right)\gg \mathrm{Card}\left(B_i\right).$$

Then, we have the following:

$$W_{ij}\rightleftarrows B_{is}, j=\mathrm{1,2},\dots ,r.$$

In this case, a separate virtual subclass room should be created. A crucial point in the education of people is that they strengthen the newly acquired knowledge by presenting results to other people. Finally, the supervisor should control the whole learning process. The deep e-learning algorithm is presented in Algorithm 1 using pseudocode.

Algorithm 1 The deep recursive e-learning algorithm

define$\mathrm{the} \mathrm{main} \mathrm{problem} P$;

decompose$\mathrm{the} \mathrm{problem} P$$\mathrm{into} \mathrm{a} \mathrm{set} \mathbf{P}=\{P_1, P_2, \dots ,P_n\}$ of elementary tasks;

$\mathrm{for} i=1$ to n do

begin

create$\mathrm{a} \mathrm{base} \mathrm{of} \mathrm{knowledge} K_i=\left\{K_{i1},K_{i2},\dots ,K_{i{\nu }_i}\right\}$$\mathrm{related} \mathrm{to} \mathrm{the} \mathrm{subproblem} P_i$;

send $P_i$ to all students

receive feedback from all students;

rank the results of the students;

choose$\mathrm{the} \mathrm{set} \mathrm{of} \mathrm{the} \mathrm{students} B_i=\left\{B_{i1}, B_{i2}, \dots ,B_{ik}\right\}$$\mathrm{with} \mathrm{a} \mathrm{full} \mathrm{solution} \mathrm{of} \mathrm{the} \mathrm{subproblem} P_i$;

mark $\mathrm{all} \mathrm{students} \mathrm{with} \mathrm{no} \mathrm{solution} \mathrm{of} P_i$ by

$W_i=\left\{W_{i1}, W_{i2}, \dots {,W}_{il}\right\}$;

determine$\mathrm{the} \mathrm{students} I_i=\left\{I_{i1},I_{i2},\dots ,I_{im}\right\}$$\mathrm{who} \mathrm{have} \mathrm{incomplete} \mathrm{solutions} \mathrm{to} \mathrm{the} \mathrm{subproblem} P_i$;

set a peer-to-peer relation $W_{ir}\rightleftarrows B_{is}$$\mathrm{where} W_{ir}$$\mathrm{asks} \mathrm{questions} \mathrm{about} P_i$$\mathrm{and} B_{is}$ answers;

$p=1$;

while $\mathrm{some} \mathrm{students} \mathrm{of} W_i$$\mathrm{do} \mathrm{not} \mathrm{have} \mathrm{a} \mathrm{solution} \mathrm{to} P_i$ do

begin

send $\mathrm{a} \mathrm{set} \mathrm{of} \mathrm{knowledge} K_{ip}\left(W_{ir}\right)=$$\{\mathrm{theorems}, \mathrm{rules}, \mathrm{properties}, \mathrm{example} \mathrm{etc}.\} \mathbf{to} W_{ir}$;

expand $K_{ip}\left(W_{ir}\right)$$\mathrm{to} K_{ip+1}\left(W_{ir}\right)$ concerning the difficulties of the students;

$p=p+1$

end;

determine $\mathrm{an} \mathrm{initial} \mathrm{value} \mathrm{of} p$$\mathrm{for} \mathrm{each} \mathrm{student} \mathrm{from} I_i$;

while $\mathrm{some} \mathrm{students} \mathrm{of} I_i$$\mathrm{do} \mathrm{not} \mathrm{have} \mathrm{a} \mathrm{solution} \mathrm{to} P_i$ do

begin

send $\mathrm{a} \mathrm{set} \mathrm{of} \mathrm{knowledge} K_{ip}\left(I_{iq}\right)=$$\{\mathrm{theorems}, \mathrm{rules}, \mathrm{properties}, \mathrm{example} \mathrm{etc}.\} \mathbf{to} I_{iq}$;

expand $K_{ip}\left(I_{iq}\right)$$\mathbf{to} K_{ip+1}\left(I_{iq}\right)$ regarding the difficulties of the students;

$p=p+1$

end;

end.

3. Model Problem

Let the domain $\mathsf{\Omega }$ be defined by

$$\mathsf{\Omega }=\{\left(x,y\right) | 1<x<s, 1<y<s\}$$

and $v$ be a square-summable function in $\mathsf{\Omega }.$ We define a quadratic operator

$$Av(p,q)=\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{{(s-1)}^2}}{\int }_1^s{\int }_1^{s}v\left(p,y\right)v\left(x,q\right).Budxdy,$$

with an operator-valued kernel [20] as follows:

$$Bu={\displaystyle \frac{u\left(x\right)-u\left(y\right)}{x-y}}.$$

Problem 1. 

By varying the kernel, the main goal is to establish whether some specific functions belong to $domA$. To demonstrate the idea, we choose $v(x,y)={\displaystyle \frac{y}{x}}$  and $u=\mathit{ln}x$. Then,

$$Av\left(p,q\right)={\displaystyle \frac{q}{p}}\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{\mathrm{meas}\mathsf{\Omega }}}{\int }_1^s{\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy.$$

Thus, it is enough to analyze the existence and boundness of the limit

$$L=\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{\mathrm{meas}\mathsf{\Omega }}}{\int }_1^s{\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy.$$

Solution Initially, we decompose the original integral into a sum of two integrals, as follows:

$$\begin{gathered} L=\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{{\left(s-1\right)}^2}}{\int }_1^s{\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy \\ =\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{{\left(s-1\right)}^2}}\left({\int }_1^s{\int }_1^y{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln y-\ln x}{y-x}}dxdy+{\int }_1^s{\int }_y^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy\right). \end{gathered}$$

The solution needs some preliminary definitions. Let

$$\begin{gathered} {\mathsf{\Omega }}_1=\{\left(x,y\right) \left|1<x<y, 1<y<s\}, {\mathsf{\Omega }}_2=\{\left(x,y\right) \right| y<x<s, 1<y<s\}, \\ \phi \left(x,y\right)={\displaystyle \frac{\ln y-\ln x}{y-x}}, f\left(x,y\right)={\displaystyle \frac{y}{x}}\phi \left(x,y\right) \end{gathered}$$

and

$$F_1\left(s\right)={\int }_1^s{\int }_1^{y}f\left(x,y\right)dxdy, F_2\left(s\right)={\int }_1^s{\int }_y^{s}f\left(x,y\right)dxdy.$$

Obviously, $\phi \left(x,y\right)>0$ when $\left(x,y\right)\in {\mathsf{\Omega }}_1\cup {\mathsf{\Omega }}_2$ and $F_i\left(s\right)>0, i=\mathrm{1,2}, \forall s>1.$ Additionally, we denote the Zhukovskii function $\sigma \left(x\right)$ by

$$\sigma \left(x\right)={\displaystyle \frac{1}{2}}\left(x+{\displaystyle \frac{1}{x}}\right).$$

To apply L’Hospital’s theorem, we have to prove that

$${\int }_1^s{\int }_1^{s}f\left(x,y\right)dxdy$$

tends to infinity when $s\to +\infty .$ To this end, we estimate $f\left(x,y\right)$ from below. First, we consider the function $F_1\left(s\right).$ We present the function

$$\phi \left(x,y\right)={\displaystyle \frac{\ln \frac{y}{x}}{y-x}}={\displaystyle \frac{\ln \left(1+z\right)}{zx}},\left(x,y\right)\in {\mathsf{\Omega }}_1$$

by a new positive variable $z={\displaystyle \frac{y-x}{x}}.$ The inequality

$${\displaystyle \frac{x}{x+1}}\le \ln \left(1+x\right)\le x, x>-1$$

(3)

leads to

$$\phi \left(x,y\right)>{\displaystyle \frac{z}{z+1}}.{\displaystyle \frac{1}{zx}}={\displaystyle \frac{1}{(z+1)x}}={\displaystyle \frac{1}{y}}.$$

Thus,

$$\phi \left(x,y\right)>{\displaystyle \frac{1}{y}} \forall \left(x,y\right)\in {\mathsf{\Omega }}_1.$$

By applying the right-hand side of (3), we obtain the following:

$${\displaystyle \frac{1}{y}}<\phi \left(x,y\right)<{\displaystyle \frac{1}{x}} \forall \left(x,y\right)\in {\mathsf{\Omega }}_1$$

(4)

and

$$F_1\left(s\right)>{\int }_1^s{\int }_1^y{\displaystyle \frac{1}{x}}dxdy=1+\mathrm{s}\left(\ln s-1\right).$$

On the contrary, we have

$${\displaystyle \frac{1}{x}}<\phi \left(x,y\right)<{\displaystyle \frac{1}{y}} \forall \left(x,y\right)\in {\mathsf{\Omega }}_2$$

and

$$F_2\left(s\right)>{\int }_1^s{\int }_y^s{\displaystyle \frac{y}{x^2}}dxdy=\xi \left(s\right)-1.$$

Then,

$$\underset{s\to +\infty }{\lim }{\int }_1^s{\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy\ge \underset{s\to +\infty }{\lim }\left(\mathrm{s}\left(\ln s-1\right)+\sigma \left(s\right)\right)=+\infty .$$

The latter result guarantees the validity of L’Hospital’s theorem applied to the initial limit

$$\begin{gathered} \underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{{\left(s-1\right)}^2}}{\int }_1^s{\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dxdy \\ ={\displaystyle \frac{1}{2}}\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{s-1}}\left({\int }_1^s{\displaystyle \frac{s}{x}}.{\displaystyle \frac{\ln x-\ln s}{x-s}}dx+{\int }_1^s{\displaystyle \frac{\partial }{\partial s}}\left({\int }_1^s{\displaystyle \frac{y}{x}}.{\displaystyle \frac{\ln x-\ln y}{x-y}}dx\right)dy\right) \\ ={\displaystyle \frac{1}{2}}\underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{s-1}}\left({\int }_1^s{\displaystyle \frac{s}{x}}.{\displaystyle \frac{\ln s-\ln x}{s-x}}dx+{\int }_1^s{\displaystyle \frac{y}{s}}.{\displaystyle \frac{\ln s-\ln y}{s-y}}dy\right) \\ ={\displaystyle \frac{1}{2}}\underset{s\to +\infty }{\lim }{\displaystyle \frac{2}{s-1}}{\int }_1^s\sigma \left({\displaystyle \frac{x}{s}}\right)\phi \left(x,s\right)dx. \end{gathered}$$

Let

$$\mathsf{\Phi }\left(s\right)={\int }_1^s\sigma \left({\displaystyle \frac{x}{s}}\right)\phi \left(x,s\right)dx.$$

There is no doubt that $\phi \left(x,s\right)>0$ when $1\le x<s$ and $\mathsf{\Phi }\left(s\right)>0, \forall s>1.$ The validity of

$${\displaystyle \frac{1}{s}}<\phi \left(x,s\right)<{\displaystyle \frac{1}{x}}, 1\le x<s$$

follows from (4). Therefore,

$$\begin{gathered} \underset{s\to +\infty }{\lim }{\displaystyle \frac{\mathsf{\Phi }\left(s\right)}{s-1}}\le \underset{s\to +\infty }{\lim }{\displaystyle \frac{1}{s-1}}{\int }_1^s{\displaystyle \frac{1}{x}} \sigma \left({\displaystyle \frac{x}{s}}\right)dx\le \underset{s\to +\infty }{\lim }{\displaystyle \frac{\sigma \left(\frac{s}{s}\right)\left(s-1\right)}{s-1}}= \\ =\underset{s\to +\infty }{\lim }\sigma \left(1\right)=1. \end{gathered}$$

Finally, $L\le 1$ and $v\in \mathrm{d}\mathrm{o}\mathrm{m}A$.

We subdivide the main problem into easily-solvable subproblems as follows.

The main problem Does $v={\displaystyle \frac{y}{x}}$ belong to $\mathrm{d}\mathrm{o}\mathrm{m}A?$

Subproblem 1 Is the argument of operator $A$ a symmetric function?

Subproblem 2 The kernel has singularity in the interior of domain $\mathsf{\Omega }.$

Subproblem 3 Separate domain $\mathsf{\Omega }$ into subdomains so that the singularities of the kernel are only on the boundaries of each subdomain.

Subproblem 4 Representation of the original integral as a sum of integrals over the subdomains.

Subproblem 5 Introduce an additional function to optimize the denotations.

Subproblem 6 Obtain double-sided estimates for the kernel. This is the crucial point for solving the main problem.

Subsubproblem 6 (a) Estimate from below.

Subsubproblem 6 (b) Estimate from above.

Note that the estimates for the kernel are not the same in both subdomains.

Subproblem 7 Prove that both integrals are positive for all $s>1.$

Subproblem 8 Investigate the behavior of the integrals over subdomains ${\mathsf{\Omega }}_1$ and ${\mathsf{\Omega }}_2$ as $s\to \infty$.

Subproblem 9 Apply L’Hospital’s theorem to analyze the limit of the integral over each subdomain as $s\to \infty$.

Subproblem 10 Apply appropriate inequalities to establish lower and upper bounds for the integrals over the subdomains.

Subproblem 11 Merge the one-dimensional integrals in a common domain and analyze their behavior when $s\to \infty$.

Subproblem 12 Estimate the integrand of the resulting one-dimensional integral.

Subproblem 13 Use the results obtained from the analysis of the sum of integrals to determine whether the function $v={\displaystyle \frac{y}{x}}$ belongs to $\mathrm{d}\mathrm{o}\mathrm{m}A$.

A discussion on specific partial cases generates an additional outcome. We consider some partial cases when the necessary overall work can be essentially reduced.

• The multiplier $v\left(p,y\right)v\left(x,q\right)$ is symmetric.
• The function $v(x,y)$ is symmetric.
• The kernel is square-summable $Bu\in L^2\left(\mathsf{\Omega }\right)$.
• The argument of the operator $B$ is differentiable in $\left[1,\infty \right).$

  (a)

  $u^{\prime }$ is monotone and can be estimated by $\psi \left(y\right)\le u^{\prime }\left(\xi \right)\le \psi \left(x\right),$

  $$\xi \in \left(y,x\right).$$

  (b)

  $u^{\prime }$ is nonmonotone but can be estimated by

  $$\psi \left(x,y\right)\le u^{\prime }\left(\xi \right)\le \psi \left(y,x\right).$$

We can use $u=\ln x$ and $u=\arctan x$ for the argument of $B$ for instance. As an inappropriate example, we indicate $u=\sqrt{x-2}.$

We continue with some examples of improper integrals that can be included in model problems.

Example 1. 

The one-dimensional weak fractional Laplacian [21,22]

$$L\left(v\right)={\displaystyle \frac{c\left(\alpha \right)}{2}}\left(\underset{D_s}{\overset{ }{\int }}\underset{D_s}{\overset{ }{\int }}\left(v\left(y\right)-v\left(x\right)\right)Bu\right.dxdy\left.+2\underset{D_s}{\overset{ }{\int }}\underset{D_s^{\perp }}{\overset{ }{\int }}\left(v\left(y\right)-v\left(x\right)\right)Budydx\right),$$

where

$$Bu={\displaystyle \frac{u\left(y\right)-u\left(x\right)}{{\left|y-x\right|}^{1+2\alpha }}}, D_s=\left[1,s\right], D_s^{\perp }=\mathbf{R}\backslash D_s,$$

$$c\left(\alpha \right)={\displaystyle \frac{4^{\alpha }\alpha \Gamma \left(\alpha +\frac{1}{2}\right)}{\sqrt{\pi }\Gamma \left(1-\alpha \right)}}, \alpha \in \left(\mathrm{0,1}\right),$$

$\Gamma \left(x\right)$ is the well-known gamma function.

The case when $s$ tends to infinity is particularly interesting. The case $\alpha ={\displaystyle \frac{1}{2}}$ could be used as a starting point.

Example 2. 

The linear functional

$$F\left(v\right)=\underset{s\to +\mathrm{\infty }}{\lim }{\displaystyle \frac{1}{{\left(s-1\right)}^2}}{\int }_1^s{\int }_1^{s}v\left(x,y\right).Budxdy.$$

Example 3. 

The Laplace transform $L$ maps the function $f\left(t\right)$ into a function

$$F\left(p\right)={\int }_0^{\infty }f\left(t\right)e^{-pt}dt.$$

Example 4. 

The Poisson-type integrals

$${\int }_0^{\infty }f\left(x\right)e^{-x^2}dx, {\int }_0^{\infty }f\left(x\right){\displaystyle \frac{\sin x}{x}} dx$$

Example 5. 

The Fresnel-type integrals

$${\int }_0^{\infty }f\left(x\right)\sin x^{2}dx, {\int }_0^{\infty }f\left(x\right)\cos x^{2}dx.$$

4. Real-Life Experiments

We conducted experiments in small well-distributed groups. The MS Teams environment was used for communication between the students and the supervisor. The electronic classroom, peer-to-peer, and client-server communications have been dynamically created in the MS Teams environment. The necessary knowledge bases had been previously created in MS Word format. The students and lecturer wrote in the same MS Word file during the discussions. The exchange of knowledge has been implemented using MS Teams. Here we have to emphasize that the ideas of the proposed theory can be successfully implemented with other e-learning systems.

Students have reported various difficulties in learning improper integrals. Most of them are related to the points of discontinuity, continuity, or differentiability of the integrand. The real-life tests indicated that students have the hardest time estimating transcendental functions and non-linear operators. The choice of suitable inequalities and the corresponding comparison criteria is a major goal in teaching improper integrals. We pay special attention to classes of divergent integrals that have finite Cauchy principal values. Other difficulties that we noticed among the students during actual tests were the operation with nested linear operators and the calculation of non-linear functionals. To this end, we developed multistage model problems to test the students’ ability to explain the operations they have performed. When applying the partition of unity method, we ensure that students motivate their actions by applying specific theoretical knowledge. Many students perform various operations and conversions intuitively, which creates hidden opportunities for further errors. Some of them often operate by analogy without considering the specific features of the integrands and domains of integration. Students have often applied definite integral properties to calculations of improper integrals without being sure that such properties hold for the improper integrals. As a typical example of our observations, we mention the Newton–Leibniz formula. Some basic requirements should be satisfied before starting the experiments:

• The students should be strongly motivated to participate in this type of teaching process;
• The compilation of a well-distributed group is essential for the successful application of the method;
• The students should have a strong background in the field of improper integrals before starting to solve multistage problems.

Experimentally, we have established the following advantages of the deep recursive e-learning process based on the partition of unity method:

• In their efforts to explain the transformations that they have carried out, the students notice various gaps in their knowledge, which the supervisor subsequently removes;
• Dividing the main problem into a large number of easily understandable subproblems significantly facilitates the formation of new skills and the construction of lasting knowledge in students;
• Working in a team by building peer-to-peer communication gives much better results than independent training under the guidance of a supervisor;
• The communication $W_{ir}\rightleftarrows B_{is}$is very important, because the students with a full solution should start the applications of knowledge from the beginning and use all possible theory $K_i$ related to the specific subproblem $P_i$;
• Students who have a partial solution are not suitable to contact either those who have a complete solution or those who do not have a solution to a specific subproblem; therefore, we propose that $I_{iq}\rightleftarrows \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{r}$.
• The teaching process is finished on this level when the students can make successful subdivisions of multistage problems.

5. Conclusions

The main goal of this research is to present a new approach to teaching students of mathematics majors, in which students consolidate newly acquired knowledge through discussions with other students. To this end, a grading procedure and a new partition of unity method are introduced. The partition of unity method makes it possible to determine the specific difficulties of each student. In this way, the necessary knowledge bases can be selectively generated for each student. The e-learning process is provided with a deep recurrent procedure, which produces an additional outcome. The new approach makes students an active part of the teaching process. Some helpful examples of improper integral operators are presented. They can be successfully used in a real-life e-learning process.