﻿ Closed Forms of Some Definite Integrals

### Closed Forms of Some Definite Integrals

Chii-Huei Yu

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## Closed Forms of Some Definite Integrals

Chii-Huei Yu

Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan

### Abstract

This paper studies two types of definite integrals and uses Maple for verification. The closed forms of these definite integrals can be obtained using Poisson integral formula. On the other hand, some examples are used to demonstrate the calculations.

• Yu, Chii-Huei. "Closed Forms of Some Definite Integrals." Automatic Control and Information Sciences 2.3 (2014): 49-52.
• Yu, C. (2014). Closed Forms of Some Definite Integrals. Automatic Control and Information Sciences, 2(3), 49-52.
• Yu, Chii-Huei. "Closed Forms of Some Definite Integrals." Automatic Control and Information Sciences 2, no. 3 (2014): 49-52.

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### 1. Introduction

In calculus and engineering mathematics, there are many methods to solve the integral problems including change of variables method, integration by parts method, partial fractions method, trigonometric substitution method, etc. In this paper, we study the following two types of definite integrals which are not easy to obtain their answers using the methods mentioned above.

 (1)
 (2)

where are real numbers, , and is a positive integer. We can obtain the closed forms of these definite integrals using Poisson integral formula; these are the major results of this paper (i.e., Theorem A). 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], Yu and Sheu [31, 32, 33], and T. -J. Chen and Yu [34, 35, 36] used some methods including complex power series method, integration term by term theorem, differentiation with respect to a parameter, Parseval’s theorem, area mean value theorem, and generalized Cauchy integral formula to solve some types of integrals. In this article, some examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.

### 2. Main Results

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

2.1. Notations
2.1.1. Let be a real number, the largest integer less than or equal to is denoted as .
2.1.2. Suppose that is a real number, then for positive integers ; .

2.2. Formulas
2.2.1. Euler’s formula

, where , and is any real number.

2.2.2. DeMoivre’s formula:

, where is any integer, and is any real number.

2.2.3. , where are real numbers.

An important formula used in this study is introduced below, which can be found in [[37], p 145].

2.2.4. Poisson integral formula:

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

2.2.5. Binomial Theorem

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

Before deriving the major results in this study, two lemmas are needed.

Lemma 1 Suppose that is a real number, and is a positive integer. Then:

 (3)

and

 (4)

Proof

Lemma 2 Assume that are real numbers, , and is a non-negative integer. Then:

 (5)
 (6)

Proof Because is analytic on the whole complex plane. Using Poisson integral formula yields:

 (7)

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

 (8)

Thus,

 (9)

By the equality of real parts of both sides of Eq. (9), we obtain Eq. (5). The equality of imaginary parts of both sides of Eq. (9) yields Eq. (6) holds.

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

Theorem A If are real numbers, , and is a positive integer, then the definite integrals:

 (10)

and

 (11)

Proof

On the other hand,

### 3. Examples

In the following, for the two types of definite integrals in this study, we provide some examples and use Theorem A to determine their closed forms. In addition, Maple is used to calculate the approximations of these definite integrals and their solutions for verifying our answers.

3.1. Example In Eq. (10), if , and , then the definite integral:

 (12)

Next, we use Maple to verify the correctness of Eq. (12).

>evalf(int((cos(theta))^5/(25-24*cos(theta-Pi/3)), theta=0..2*Pi),18);

0.0986952281919993812

>evalf(3603*Pi/114688,18);

0.0986952281919993812

On the other hand, let , and in Eq. (10), then we obtain:

 .(13)

We also use Maple to verify the correctness of Eq. (13).

>evalf(int((cos(theta))^4/(61-60*cos(theta-Pi/4)), theta=0..2*Pi),18);

0.179766709256865449

>evalf(3263*Pi/57024,18);

0.179766709256865449

3.2. Example In Eq. (11), if , and , then the definite integral:

 (14)

Using Maple to verify the correctness of Eq. (14) as follows:

>evalf(int((sin(theta))^3/(13+12*cos(theta+Pi/6)),theta=0..2*Pi),18);

0.221075038585948413

>evalf(19*Pi/270,18);

0.221075038585948413

In addition, let , and in Eq. (11), then:

 (15)

>evalf(int((sin(theta))^8/(25+24*cos(theta-3*Pi/4)), theta=0..2*Pi),18);

0.184012744327370238

>evalf(1719713*Pi/29360128,18);

0.184012744327370238

### 4. Conclusion

In this paper, we use Poisson integral formula to solve two types of 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. On the other hand, 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.

### 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] C. -H. Yu and S. -D. Sheu, “Infinite series forms of double integrals,” International Journal of Data Envelopment Analysis and *Operations Research*, Vol. 1, No. 2, pp. 16-20, 2014. In article [32] C. -H. Yu and S. -D. Sheu, “Using area mean value theorem to solve some double integrals,” Turkish Journal of Analysis and Number Theory, Vol. 2, No. 3, pp. 75-79, 2014. In article [33] C. -H. Yu and S. -D. Sheu, “Evaluation of triple integrals,” American Journal of Systems and Software, Vol. 2, No. 4, pp. 85-88, 2014. In article [34] 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 [35] 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 [36] 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 [37] J. E. Marsden, Basic complex analysis, W. H. Freeman and Company, San Francisco, 1973. In article
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