Evaluating Multiple Integrals Using Maple

This paper uses the mathematical software Maple for the auxiliary tool to study two types of multiple integrals. We can obtain the infinite series forms of these two types of multiple integrals by using binomial series and integration term by term theorem. On the other hand, 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.


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 Mozart, 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.
The multiple integral problem is closely related with probability theory and quantum field theory, and can refer to [8][9]. For this reason, the evaluation and numerical calculation of multiple integrals is important. In this paper, we mainly study the following two types of n -tuple integrals Where n is any positive integer, are real numbers for all n k ,.., 1 = . We can obtain the infinite series forms of these two types of multiple integrals by using binomial series and integration term by term theorem; these are the major results of this study (i.e., Theorems 1 and 2). Moreover, we obtain some corollaries from these two theorems. For the study of related multiple integral problems can refer to [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. In addition, we provide some multiple 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. Therefore, Maple provides insights and guidance regarding problem-solving methods.

Main Results
Firstly, we introduce two notations and two important theorems used in this paper. The following is the first result in this study, we find the infinite series forms of the multiple integral (1).

Theorem 1
Suppose n is any positive integer, and . Then the n -tuple integral  ,.., Therefore, we obtain the n -tuple integral Thus, the n -tuple integral By Theorem1, we immediately have the following result.

Corollary 1 Suppose n is any positive integer
are real numbers for all n k ,.., Then the n -tuple improper integral The following is the second major result in this paper, we determine the infinite series form of the multiple integral (2).

Theorem 2
Therefore, we obtain the n -tuple integral Thus, we obtain the n -tuple integral q.e.d. By Theorem 2, we obtain the following result.

Examples
In the following, for the two types of multiple integrals in this study, we provide some examples and use our theorems and corollaries to determine the infinite series forms of these multiple integrals. On the other hand, we employ Maple to calculate the approximations of these multiple integrals and their solutions for verifying our answers.