1. Introduction

The computer algebra system (CAS) has been widely employed in mathematical and scientific studies. The rapid computations and the visually appealing graphical interface of the program render creative research possible. Maple possesses significance among mathematical calculation systems and can be considered a leading tool in the CAS field. The superiority of Maple lies in its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. In addition, through the numerical and symbolic computations performed by Maple, the logic of thinking can be converted into a series of instructions. The computation results of Maple can be used to modify our previous thinking directions, thereby forming direct and constructive feedback that can aid in improving understanding of problems and cultivating research interests.

(1)

(2)

(3)

(4)

where are real numbers, , and are positive integers. The closed forms of these definite integrals can be obtained mainly using Poisson integral formula; these are the major results of this paper (i.e., Theorems 1 and 2). Adams et al. ^[1], Nyblom ^[2], and Oster ^[3] provided some techniques to solve the integral problems. Yu [4-29]^[4], Yu and B. -H. Chen ^[30], and T. -J. Chen and Yu ^{[31, 32, 33]} used complex power series method, integration term by term theorem, differentiation with respect to a parameter, Parseval’s theorem, and generalized Cauchy integral formula to solve some types of integrals. In this paper, some examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.

2. Preliminaries and Main Results

Some notations and formulas used in this paper are introduced below.

Suppose that is a real number, and is a positive integer. Define , and .

, where , and is any real number.

, where is an integer, and is a real number.

The following is an important formula used in this study, which can be found in [^[34], p 145].

Suppose that are real numbers, and . If is defined and continuous on the closed disc and is analytic on the open disc , then

, where are complex numbers, and is a positive integer.

In the following, we determine the closed forms of the definite integrals (1) and (2).

Theorem 1. If are real numbers, , and are positive integers, then the definite integrals

(5)

and

(6)

Proof Let , then is defined and continuous on the closed disc, and it is analytic on the open disc . Let , then using Poisson integral formula for yields

(7)

By Euler’s formula and DeMoivre’s formula, we have

(8)

It follows that

(9)

Using binomial theorem yields

(10)

By the equality of real parts of both sides of Eq. (10), we obtain Eq. (5). Also, using the equality of imaginary parts of both sides of Eq. (10) yields Eq. (6) holds.

Next, the closed forms of the definite integrals (3) and (4) can be obtained below.

Theorem 2. Let be real numbers, , and be a positive integer, then the definite integrals

(11)

and

(12)

Proof Let , then is defined and continuous on the closed disc and it is analytic on the open disc . If , then by Poisson integral formula for we obtain

(13)

It follows that

(14)

By binomial theorem, we have

(15)

Using the equality of real parts of both sides of Eq. (15) yields Eq. (11) holds. Also, Eq. (12) can be obtained using the equality of imaginary parts of both sides of Eq. (15).

3. Examples

In the following, for the four types of definite integrals in this study, we provide two examples and use Theorems 1 and 2 to obtain their closed forms. In addition, Maple is used to calculate the approximations of these definite integrals and their solutions for verifying our answers.

In Eq. (5), if , , and , then the definite integral

(16)

Next, we use Maple to verify the correctness of Eq. (16).

>evalf(int((16*cos(4*theta-2*Pi/3)+24*cos(3*theta-Pi/3)+9*cos(2*theta))/((13-12*cos(theta-Pi/6))*(25+24*cos(theta-Pi/3))^2),theta=0..2*Pi),18);

0.00689858296836092382

>evalf((36+16*sqrt(3))*Pi/(14985+8100*sqrt(3)),18);

0.00689858296836092381

On the other hand, if , , and in Eq. (6), then

(17)

We also use Maple to verify the correctness of Eq. (17).

>evalf(int((-216*sin(5*theta-3*Pi/4)-540*sin(4*theta-Pi/2)-450*sin(3*theta-Pi/4)-125*sin(2*theta))/ ((34+30*cos(theta+Pi/3))*(61+60*cos(theta-i/4))^3), theta=0..2*Pi),18);

-0.0000443409234263192035

>evalf(-243*Pi/25000*(216*sin(5*Pi/12)+162-81*sqrt(2)-27/2*sqrt(3))/(45-36*cos(7*Pi/12))^3,18);

-0.0000443409234263192039

In Eq. (11), if , and , then

(18)

Using Maple to verify the correctness of Eq. (18) as follows:

>evalf(int(exp(12*cos(theta+Pi/4))*(-7*cos(12*sin (theta+Pi/4)+Pi/4)+6*cos(12*sin(theta+Pi/4)-theta))/ ((52+48*cos(theta-3*Pi/4))*(85-4*cos(theta+Pi/4))), theta=0..2*Pi),18);

-220.734122999910046

>evalf(-exp(8)*sqrt(2)*Pi/60,18);

-220.734122999910047

Also, let , and in Eq. (12), then the definite integral

(19)

>evalf(int(exp(-24*cos(theta-Pi/6))*(9*sin(-24*sin(theta-Pi/6)-2*Pi/3)-8*sin(-24*sin(theta-Pi/6)-theta))/((73+48*cos(theta+Pi/4))*(145-144*cos(theta-2*Pi/3))),theta=0..2*Pi),18);

0.1801875656110764

>evalf(2*Pi/55*(exp(9*cos(5*Pi/12))*(9*sin(-9*sin(5*Pi/12)-2*Pi/3)+3*sin(-9*sin(5*Pi/12)+Pi/4)))/ (90+54*cos(11*Pi/12)),16);

0.1801875656110781

4. Conclusion

In this study, we mainly use Poisson integral formula to solve some definite integrals. In fact, the applications of this formula 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 use Maple to verify our answers. These results will be used as teaching materials for Maple on education and research to enhance the connotations of calculus and engineering mathematics.

References

[1]	A. A. Adams, H. Gottliebsen, S. A. Linton, and U. Martin, “Automated theorem proving in support of computer algebra: symbolic definite integration as a case study,” Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, Canada, pp. 253-260, 1999.
	In article		CrossRef
[2]	M. A. Nyblom, “On the evaluation of a definite integral involving nested square root functions,” Rocky Mountain Journal of Mathematics, Vol. 37, No. 4, pp. 1301-1304, 2007.
	In article		CrossRef
[3]	C. Oster, “Limit of a definite integral,” SIAM Review, Vol. 33, No. 1, pp. 115-116, 1991.
	In article		CrossRef
[4]	C. -H. Yu, “A study of two types of definite integrals with Maple,” Jökull Journal, Vol. 64, No. 2, pp. 543-550, 2014.
	In article
[5]	C. -H. Yu, “Evaluating two types of definite integrals using Parseval’s theorem,” Wulfenia Journal, Vol. 21, No. 2, pp. 24-32, 2014.
	In article
[6]	C. -H. Yu, “Solving some definite integrals using Parseval’s theorem,” American Journal of Numerical Analysis, Vol. 2, No. 2, pp. 60-64, 2014.
	In article
[7]	C. -H. Yu, “Some types of integral problems,” American Journal of Systems and Software, Vol. 2, No. 1, pp. 22-26, 2014.
	In article
[8]	C. -H. Yu, “Using Maple to study the double integral problems,” Applied and Computational Mathematics, Vol. 2, No. 2, pp. 28-31, 2013.
	In article		CrossRef
[9]	C. -H. Yu, “A study on double Integrals,” International Journal of Research in Information Technology, Vol. 1, Issue. 8, pp. 24-31, 2013.
	In article
[10]	C. -H. Yu, “Application of Parseval’s theorem on evaluating some definite integrals,” Turkish Journal of Analysis and Number Theory, Vol. 2, No. 1, pp. 1-5, 2014.
	In article
[11]	C. -H. Yu, “Evaluation of two types of integrals using Maple,” Universal Journal of Applied Science, Vol. 2, No. 2, pp. 39-46, 2014.
	In article
[12]	C. -H. Yu, “Studying three types of integrals with Maple,” American Journal of Computing Research Repository, Vol. 2, No. 1, pp. 19-21, 2014.
	In article
[13]	C. -H. Yu, “The application of Parseval’s theorem to integral problems,” Applied Mathematics and Physics, Vol. 2, No. 1, pp. 4-9, 2014.
	In article
[14]	C. -H. Yu, “A study of some integral problems using Maple,” Mathematics and Statistics, Vol. 2, No. 1, pp. 1-5, 2014.
	In article
[15]	C. -H. Yu, “Solving some definite integrals by using Maple,” World Journal of Computer Application and Technology, Vol. 2, No. 3, pp. 61-65, 2014.
	In article
[16]	C. -H. Yu, “Using Maple to study two types of integrals,” International Journal of Research in Computer Applications and Robotics, Vol. 1, Issue. 4, pp. 14-22, 2013.
	In article
[17]	C. -H. Yu, “Solving some integrals with Maple,” International Journal of Research in Aeronautical and Mechanical Engineering, Vol. 1, Issue. 3, pp. 29-35, 2013.
	In article
[18]	C. -H. Yu, “A study on integral problems by using Maple,” International Journal of Advanced Research in Computer Science and Software Engineering, Vol. 3, Issue. 7, pp. 41-46, 2013.
	In article
[19]	C. -H. Yu, “Evaluating some integrals with Maple,” International Journal of Computer Science and Mobile Computing, Vol. 2, Issue. 7, pp. 66-71, 2013.
	In article
[20]	C. -H. Yu, “Application of Maple on evaluation of definite integrals,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 823-827, 2013.
	In article
[21]	C. -H. Yu, “Application of Maple on the integral problems,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 849-854, 2013.
	In article
[22]	C. -H. Yu, “Using Maple to study the integrals of trigonometric functions,” Proceedings of the 6th IEEE/International Conference on Advanced Infocomm Technology, Taiwan, No. 00294, 2013.
	In article
[23]	C. -H. Yu, “A study of the integrals of trigonometric functions with Maple,” Proceedings of the Institute of Industrial Engineers Asian Conference 2013, Taiwan, Springer, Vol. 1, pp. 603-610, 2013.
	In article		CrossRef
[24]	C. -H. Yu, “Application of Maple on the integral problem of some type of rational functions,” (in Chinese) Proceedings of the Annual Meeting and Academic Conference for Association of IE, Taiwan, D357-D362, 2012.
	In article
[25]	C. -H. Yu, “Application of Maple on some integral problems,” (in Chinese) Proceedings of the International Conference on Safety & Security Management and Engineering Technology 2012, Taiwan, pp. 290-294, 2012.
	In article
[26]	C. -H. Yu, “Application of Maple on some type of integral problem,” (in Chinese) Proceedings of the Ubiquitous-Home Conference 2012, Taiwan, pp. 206-210, 2012.
	In article
[27]	C. -H. Yu, “Application of Maple on evaluating the closed forms of two types of integrals,” (in Chinese) Proceedings of the 17th Mobile Computing Workshop, Taiwan, ID16, 2012.
	In article
[28]	C. -H. Yu, “Application of Maple: taking two special integral problems as examples,” (in Chinese) Proceedings of the 8th International Conference on Knowledge Community, Taiwan, pp. 803-811, 2012.
	In article
[29]	C. -H. Yu, “Evaluating some types of definite integrals,” American Journal of Software Engineering, Vol. 2, Issue. 1, pp. 13-15, 2014.
	In article
[30]	C. -H. Yu and B. -H. Chen, “Solving some types of integrals using Maple,” Universal Journal of Computational Mathematics, Vol. 2, No. 3, pp. 39-47, 2014.
	In article
[31]	T. -J. Chen and C. -H. Yu, “A study on the integral problems of trigonometric functions using two methods,” Wulfenia Journal, Vol. 21, No. 4, pp. 76-86, 2014.
	In article
[32]	T. -J. Chen and C. -H. Yu, “Fourier series expansions of some definite integrals,” Sylwan Journal, Vol. 158, Issue. 5, pp. 124-131, 2014.
	In article
[33]	T. -J. Chen and C. -H. Yu, “Evaluating some definite integrals using generalized Cauchy integral formula,” Mitteilungen Klosterneuburg, Vol. 64, Issue. 5, pp. 52-63, 2014.
	In article
[34]	J. E. Marsden, Basic complex analysis, W. H. Freeman and Company, San Francisco, 1973.
	In article