Solving Some Types of Integrals Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study six types of integrals. We can obtain the infinite series forms of these six types of integrals by using integration term by term theorem. In addition, we propose some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. Therefore, Maple provides insights and guidance regarding problem-solving methods. Keyword sIntegrals, Infinite Series Forms, Integration Term By Term Theorem, Maple


Introduction
As information technology advances, whether computers can become comparable with human brains to perform abstract tasks, such as abstract art similar to the paintings of Picasso and musical compositions similar to those of Beethoven, is a natural question. Currently, this appears unattainable. In addition, whether computers can solve abstract and difficult mathematical problems and develop abstract mathematical theories such as those of mathematicians also appears unfeasible. Nevertheless, in seeking for alternatives, we can study what assistance mathematical software can provide. This study introduces how to conduct mathematical research using the mathematical software Maple. The main reasons of using Maple in this study are its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. By employing the powerful computing capabilities of Maple, difficult problems can be easily solved. Even when Maple cannot determine the solution, problem-solving hints can be identified and inferred from the approximate values calculated and solutions to similar problems, as determined by Maple. For this reason, Maple can provide insights into scientific research. Inquiring through an online support system provided by Maple or browsing the Maple website (www.maplesoft.com) can facilitate further understanding of Maple and might provide unexpected insights. For the instructions and operations of Maple, [1][2][3][4][5][6][7] can be adopted as references.
In calculus and engineering mathematics courses, we learnt many methods to solve the integral problems including change of variables method, integration by parts method, partial fractions method, trigonometric substitution method, and so on.In this paper, we study the following six types of integrals which are not easy to obtain their answers using the methods mentioned above.
are real numbers? We can obtain the infinite series forms of these six types of integrals by using integration term by term theorem; these are the major results of this paper (i.e., Theorems1-6). The study of related integral problems can refer to [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. On the other hand, we provide some integrals to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

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Solving Some Types of Integrals Using Maple

Main Results
Firstly, we introduce some formulas used in this study. ,where y is a real number, 1 ≤ y . ,where y is a real number, 1 ≤ y . ,where y is a real number, 1 ≥ y .

The Inverse Secant Function
Next, we introduce an important theorem used in this paper.

Integration Term by TermTheorem ([24])
Suppose is a sequence of Lebesgue integrable functions defined on an interval . If is convergent, then .
The following is the first result of this study, we obtain the infinite series form of the integral (1).

Theorem 1
is not a non-negative integer, and C is a constant. If Next, we determine the infinite series form of the integral (2).

Theorem 2
Let the assumptions be the same as Theorem 1. q.e.d.

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Solving Some Types of Integrals Using Maple The following is the third major result in this study, we obtain the infinite series form of the integral (3).

Theorem 3
Suppose the assumptions are the same as Theorem Next, we find the infinite series form of the integral (4).

Theorem 4
Let the assumptions be the same as Theorem  Finally, we determine the infinite series form of the integral (6).

Theorem 6
Let the assumptions be the same as Theorem 5.  q.e.d.

Examples
In the following, for the six types of integrals in this study, we provide some integrals and use our theorems to determine their infinite series forms. In addition, we evaluate some related definite integrals and employ Maple to calculate the approximations of these definite integrals and their solutions for verifying our answers.

Conclusion
In this study, we provide a new technique to solve some types of integrals, and we hope this technique can be applied to evaluate another integral problem. On the other hand, we know the integration term by term theorem plays a significant role in the theoretical inferences of this study. In fact, the applications of this theorem are extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications. In addition, Maple also plays a vital assistive role in problem-solving. In the future, we will extend the research topic to other calculus and engineering mathematics problems and solve these problems by using Maple. These results will be used as teaching materials for Maple on education and research to enhance the connotations of calculus and engineering mathematics.