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Methods for Evaluating Some Integrations Involving Inverse Trigonometric Functions

Chii-Huei Yu

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

Abstract: This paper uses the mathematical software Maple as the auxiliary tool to study the integral problems. The infinite series expressions of some types of indefinite integrals involving the powers of inverse trigonometric functions can be obtained by using power series expansions, binomial series and integration term by term theorem. Moreover, some examples are proposed to demonstrate the calculations. The research approach adopted in this study is to get the answers through manual calculations and then use Maple to verify the answers. This kind of research method not only allows us to find the calculation errors, but also helps us to modify the direction of original thinking from manual calculation and Maple calculation. Therefore, Maple provides us insights into the problems.

Keywords: infinite series expressions, indefinite integrals, inverse trigonometric functions, power series expansions, binomial series, integration term by term theorem, Maple

Citation: Chii-Huei Yu, Methods for Evaluating Some Integrations Involving Inverse Trigonometric Functions. International Transaction of Electrical and Computer Engineers System, 4 (1). 8-13. January 2017. doi:10.12691/iteces-4-1-2

1. Introduction

This paper describes how to use Maple to study mathematics. The main reasons of using Maple in this article are its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. Using 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. The following is a brief chart to demonstrate our research method:

Figure 1. A chart of the research method used in this paper

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. This article considers the following six types of indefinite integrals involving the powers of inverse trigonometric functions which are not easy to obtain their answers using the methods mentioned above.

[picture] | (1)

[picture] | (2)

[picture] | (3)

[picture] | (4)

[picture] | (5)

[picture] | (6)

where are real numbers. We can obtain the infinite series expressions of these types of indefinite integrals by using power series expansions, binomial series and integration term by term theorem; these are the major results of this paper (i.e., Theorems 1-6). Adams et al. [1], Nyblom [2], and Oster [3] provided some methods to solve some integral problems. Yu [4-27], Yu and Huang [28], Yu and Chen [29], and Yu and Sheu [30,31,32] used some techniques which include complex power series expansions, integration term by term theorem, differentiation with respect to a parameter, Parseval’s theorem, and area mean value theorem to evaluate 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. Preliminaries and Results

Firstly, the power series expansions of six basic trigonometric functions are introduced below, which can be found in [[33], p44].

2.1. Formulas

2.1.1. for any real number .

2.1.2. for any real number .

2.1.3. for , where is the -th Bernoulli number.

2.1.4.

2.1.5. for , where is the -th Euler number.

2.1.6. for .

Next, we introduce two important theorems used in this study, which can be found in ([[34], p269]) and ([[34], p244]) respectively.

2.2. Integration Term By Term Theorem

Let be a sequence of Lebesgue integrable functions defined on . If is convergent, then .

2.3. Binomial Series

where are real numbers, , for any positive integer , and

In the following, we determine the infinite series expressions of the indefinite integrals (1) and (2) respectively.

Theorem 1 Suppose that are real numbers, is not a non-negative integer, , is a constant, and exists, then

[picture] | (7)

Proof

(by integration term by term theorem)

q.e.d.

Theorem 2 If are real numbers, is not a non-negative integer, , is a constant, and exists, then

[picture] | (8)

Proof

(by Formula 2.1.1)

(by integration term by term theorem)

q.e.d.

Next, the indefinite integrals (3) and (4) can be obtained as the following two theorems respectively.

Theorem 3 If are real numbers, is not a positive integer, and exists, then

[picture] | (9)

Proof

(by Formula 2.1.3)

q.e.d.

Theorem 4 If are real numbers, is not a non-negative integer, and exists, then

[picture] | (10)

Proof

(by Formula 2.1.4)

q.e.d.

Finally, we evaluate the indefinite integrals (5) and (6) below.

Theorem 5 Assume that are real numbers, , and exists.

Case 1. If and is not a non-negative integer, then

[picture] | (11)

Case 2. If , then

[picture] | (12)

Proof Case 1. If , then

(by Formula 2.1.5)

Case 2. If , then

(by binomial series)

(by Case 1)

q.e.d.

Theorem 6 Suppose that are real numbers, is not a non-negative integer, , and exists.

Case 1. If , then

[picture] | (13)

Case 2. If and exists, then

[picture] | (14)

Proof Case 1. If , then

(by Formula 2.1.6)

(by integration term by term theorem)

Case 2. If and exists, then

(by Case 1)

q.e.d.

3. Examples

In the following, for the six types of indefinite integrals in this article, some examples are proposed and we use Theorems 1-6 to determine their infinite series expressions. In addition, Maple is used to calculate the approximations of some definite integrals and their solutions for verifying our answers.

Example 1 By Theorem 1, we obtain

[picture] | (15)

for all .

Hence, the definite integral

[picture] | (16)

Using Maple to verify the correctness of Eq. (16) as follows:

>evalf(int((arcsin(x))^(2/3),x=1/2..4/5),18);

0.238813256307957296

>evalf(sum((-1)^k/((2*k)!*(2*k+5/3))*((arcsin(4/5))^(2*

k+5/3)-(arcsin(1/2))^(2*k+5/3)),k=0..infinity),18);

0.238813256307957295

On the other hand, using Theorem 2 yields

[picture] | (17)

for all .

Thus,

[picture] | (18)

Next, we use Maple to verify the correctness of Eq. (18).

>evalf(int((arccos(x))^(4/7),x=3/8..5/6),18);

0.432557196501370483

>evalf(-sum((-1)^k/((2*k+1)!*(2*k+18/7))*((arccos(5/6))

^(2*k+18/7)-(arccos(3/8))^(2*k+18/7)),k=0..infinity),18);

0.432557196501370486

Example 2 We can obtain the following result by Theorem 3,

[picture] | (19)

for all real number .

Therefore,

[picture] | (20)

Verifying the correctness of Eq. (20) using Maple.

>evalf(int((arctan(x))^(7/6),x=4..9),18);

7.46752708175224704

>evalf(sum((-1)^(k-1)*(2*k-1)*2^(2*k)*(2^(2*k)-1)*

bernoulli(2*k)/((2*k)!*(2*k+1/6))*((arctan(9))^(2*k+1/6)-(arctan(4))^(2*k+1/6)),k=1..infinity),18);

7.46752708175224684

In addition, by Theorem 4 we have

[picture] | (21)

for all real number .

Thus, the definite integral

[picture] | (22)

>evalf(int((arccot(x))^(5/9),x=3..7),18);

1.66130019704229951

>evalf(sum((-1)^k*(2*k-1)*2^(2*k)*bernoulli(2*k)/((2*

k)!*(2*k-4/9))*((arccot(7))^(2*k-4/9)-(arccot(3))^(2*k-4/9)),k=0..infinity),18);

1.66130019704229951

Example 3 The following indefinite integral can be obtained by using Eq.(12) of Theorem 5,

[picture] | (23)

for all .

Hence,

[picture] | (24)

>evalf(int((arcsec(x))^(8/3),x=-5..-2),18);

16.2861270813272605

>evalf(Pi^(8/3)*(3-sum(sum((-1)^k*(2*k)*euler(2*k)*

product(8/3-j,j=0..(m-1))/(m!*(2*k)!*(2*k+m))*(-1/Pi)^

m*((arcsec(2))^(2*k+m)-(arcsec(5))^(2*k+m)),k=0..

infinity),m=1..infinity)),18);

16.2861270813272616

On the other hand, using Eq. (14) of Theorem 6 yields

[picture] | (25)

for all .

Thus, we can determine the definite integral

[picture] | (26)

>evalf(int((arccsc(x))^6,x=-9..-2),24);

0.007590619720516328629

>evalf(-sum((-1)^(k+1)*2*(2*k-1)*(2^(2*k-1)-1)*

bernoulli(2*k)/((2*k)!*(2*k+5))*((arccsc(2))^(2*k+5)-(arccsc(9))^(2*k+5)),k=0..infinity),24);

0.007590619720516328635.

4. Conclusion

In this paper, we solve the indefinite integrals of the powers of inverse trigonometric functions. On the other hand, power series expansions, binomial series and integration term by term theorem play significant roles in the theoretical inferences of this article. In fact, the applications of the three methods 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 mathematical problems and solve these problems by using Maple. These results will be used as teaching materials for Maple on education and research to enrich the connotation 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, July 28- 31, 1999.
[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.
[3]   | C. Oster, “Limit of a definite integral,” SIAM Review, Vol. 33, No. 1, pp. 115-116, 1991.
[4]   | C. -H. Yu, “Application of complex integral on solving some integral problems of trigonometric functions,” International Journal of Research, Vol. 3, Issue. 14, pp. 4663-4668, 2016.
[5]   | C. -H. Yu, “Integral problems of trigonometric functions,” International Journal of Scientific Research in Science and Technology, Vol. 2, Issue. 1, pp. 63-67, 2016.
[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.
[7]   | C. -H. Yu,“Some types of integral problems,” American Journal of Systems and Software, Vol. 2, No. 1, pp. 22-26, 2014.
[8]   | C. -H. Yu, “Using Maple to study the double integral problems,” Applied and Computational Mathematics, Vol. 2, No. 2, pp. 28-31, 2013.
[9]   | 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, July 6-9, 2013.
[10]   | C. -H. Yu, “A study on double Integrals,” International Journal of Research in Information Technology, Vol. 1, Issue. 8, pp. 24-31, 2013.
[11]   | 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.
[12]   | C. -H. Yu, “Evaluation of two types of integrals using Maple,” Universal Journal of Applied Science, Vol. 2, No. 2, pp. 39-46, 2014.
[13]   | C. -H. Yu, “Studying three types of integrals with Maple,” American Journal of Computing Research Repository, Vol. 2, No. 1, pp. 19-21, 2014.
[14]   | C. -H. Yu, “The application of Parseval’s theorem to integral problems,” Applied Mathematics and Physics, Vol. 2, No. 1, pp. 4-9, 2014.
[15]   | C. -H. Yu, “A study of some integral problems using Maple,” Mathematics and Statistics, Vol. 2, No. 1, pp. 1-5, 2014.
[16]   | 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.
[17]   | 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.
[18]   | 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, October 25, 2012.
[19]   | 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.
[20]   | 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, December 15, 2012.
[21]   | 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.
[22]   | 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, May 31, 2012.
[23]   | C. -H. Yu, “Evaluating some integrals with Maple,” International Journal of Computer Science and Mobile Computing, Vol. 2, Issue. 7, pp. 66-71, 2013.
[24]   | C. -H. Yu, “Application of Maple on evaluation of definite integrals,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 823-827, 2013.
[25]   | 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.
[26]   | C. -H. Yu, “Application of Maple on the integral problems,” Applied Mechanics and Materials, Vols. 479-480 (2014), pp. 849-854, 2013.
[27]   | C. -H. Yu, “Evaluating some types of definite integrals,” American Journal of Software Engineering, Vol. 2, Issue. 1, pp. 13-15, 2014.
[28]   | C. -H. Yu and S. -Y. Huang, “Using Maple to evaluate two types of special integrals,” International Journal of Scientific Research in Computer Science, Engineering and Information Technology, Vol. 1, Issue. 3, pp. 1-5, 2016.
[29]   | 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.
[30]   | 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.
[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.
[32]   | 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.
[33]   | D. Zwillinger, CRC Standard Mathematical Tables and Formulae, 31st ed., Chapman & Hall/CRC, New York, 2003.
[34]   | T. M. Apostol, Mathematical Analysis, 2nd ed., Addison-Wesley, Boston, 1975.