A Study of Some Integral Problems Using Maple

This paper takes the mathematical software Maple as the auxiliary tool to study four types of integrals. We can obtain the Fourier series expansions of these four types of integrals by using integration term by term theorem. On the other hand, we provide two 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
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 mainly study the following four types of integrals which are not easy to obtain their answers using the methods mentioned above.
Where b a r , , are real numbers, 0 ≠ a . We can obtain the Fourier series expansions of these four types of integrals by using integration term by term theorem; these are the major results of this paper (i.e., Theorems 1,2). As for the study of related integral problems can refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. On the other hand, we propose 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.

Main Results
Firstly, we introduce a notation and some formulas used in this study.

Notation
Let be a complex number, where 1 − = i , are real numbers. We denote the real part of by , and the imaginary part of by .

Formulas
Im(z Next, we introduce an important theorem used in this paper.

Integration term by term theorem ([17, p269])
The following is the first major result of this study, we obtain the Fourier series expansions of the integrals (1) and (2).

Remark 1
In Theorem 1, because for each R x ∈ , It follows that we can use integration term by term theorem to show that (5) holds. The same reason that we can prove (6) by using integration term by term theorem.
Next, we determine the Fourier series expansions of the integrals (3) and (4).

Theorem 2
If the assumptions are the same as Theorem 1. Then there Mathematics and Statistics 2(  (11) Thus, for all R x ∈ , the integral ∫ + +  (12) By integration term by term theorem, it follows that for all R x ∈ , the integral C is some constant. q.e.d.

Remark 2
In Theorem 2, the reason that we can use integration term by term theorem to prove (9) and (10) is the same as Remark 1.

Examples
In the following, for the four types of integrals in this study, we propose some integrals and use Theorems 1, 2 to determine their Fourier series expansions. In addition, we evaluate some definite integrals and employ Maple to calculate the approximations of these definite integrals and their solutions for verifying our answers.