1. 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 study the following six types of definite integrals which are not easy to obtain their answers using the methods mentioned above.

(1)

(2)

(3)

(4)

(5)

(6)

where is a real number. We can obtain the infinite series forms of these definite integrals by using Parseval’s theorem; these are the major results of this paper (i.e., Theorems 1 and Theorems 2). The study of related integral problems can refer to [1-26]^[1]. On the other hand, we provide some definite 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.

2. Main Results

Firstly, we introduce a notation and a definition and some formulas used in this article.

Let be a complex number, where , are real numbers. We denote the real part of by , and the imaginary part of by .

Suppose is a continuous function defined on, the Fourier series expansion of is, where and, for all positive integers.

, where is any real number.

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

, where are real numbers.

, where are real numbers.

, where is a complex number, and are Bernoulli numbers for all positive integers .

, where is a complex number, .

Next, we introduce an important theorem used in this study.

If is a continuous function defined on , and . Suppose the Fourier series expansion of , then .

Before deriving the first major result of this paper, we need a lemma.

Suppose are real numbers with . Then

(7)

(8)

Proof

(By Formulas 2.3.3 and 2.3.4)

And

In the following, we find the infinite series forms of the definite integrals (1), (2) and (3).

Suppose is a real number with . Then the definite integrals

(9)

(10)

(11)

Proof Because

(By Formula 2.3.5)

(By DeMoivre’s formula)

(12)

(By Euler’s formula)

By Parseval’s theorem, we obtain

Similarly, because

(By Formula 2.3.5)

(13)

Also using Parseval’s theorem, we have

On the other hand, from the summation of Eq. (9) and (10) and using Eq. (8), we obtain

Before deriving the second major result of this study, we also need a lemma.

Suppose are real numbers with . Then

(14)

(15)

Proof

And

In the following, we determine the infinite series forms of the definite integrals (4), (5) and (6).

Suppose is a real number with . Then the definite integrals

(16)

(17)

(18)

Proof Because

(By Formula 2.3.6)

(19)

Using Parseval’s theorem, we have

Similarly, because

(By Formula 2.3.6)

(20)

Also by Parseval’s theorem, we obtain

In addition, from the summation of Eq. (16) and (17) and using (15), we have

3. Examples

In the following, for the definite integrals in this study, we provide some examples and use Theorems 1 and 2 to determine their infinite series forms. On the other hand, we employ Maple to calculate the approximations of these definite integrals and their solutions for verifying our answers.

Taking into Eq. (9), we obtain the definite integral

(21)

Next, we use Maple to verify the correctness of Eq. (21).

>evalf(int((sinh(1/3*cos(x))*cosh(1/3*cos(x)))^2/((sinh(1/3*cos(x)))^2+(cos(1/3*sin(x)))^2)^2,x=0..2*Pi),18);

0.349545626476568261

>evalf(Pi*sum(2^(4*n)*(2^(2*n)-1)^2*(bernoulli(2*n))^2/((2*n)!)^2*(1/3)^(4*n-2),n=1..infinity),18);

0.349545626476568260

Similarly, if in Eq. (10), we have

(22)

>evalf(int((sin(1/sqrt(2)*sin(x))*cos(1/sqrt(2)*sin(x)))^2/((sinh(1/sqrt(2)*cos(x)))^2+(cos(1/sqrt(2)*sin(x)))^2)^2,x=0..2*Pi),18);

1.61624943295020547

>evalf(Pi*sum(2^(4*n)*(2^(2*n)-1)^2*(bernoulli(2*n))^2/((2*n)!)^2*(1/sqrt(2))^(4*n-2),n=1..infinity),18);

1.61624943295020547

Finally, let in Eq. (11), then

(23)

>evalf(int(((sinh(3/4*cos(x)))^2+(sin(3/4*sin(x)))^2)/((sinh(3/4*cos(x)))^2+(cos(3/4*sin(x)))^2),x=0..2*Pi),18);

3.66517840220898049

>evalf(2*Pi*sum(2^(4*n)*(2^(2*n)-1)^2*(bernoulli(2*n))^2/((2*n)!)^2*(3/4)^(4*n-2),n=1..infinity),18);

3.66517840220898048

Let in Eq. (16), we obtain the definite integral

(24)

>evalf(int((sinh(3*cos(x))*cosh(3*cos(x)))^2/((sinh(3*cos(x)))^2+(sin(3*sin(x)))^2)^2,x=0..2*Pi),18);

11.5167959003610174

>evalf(Pi*(16/9+sum(2^(4*n)*(bernoulli(2*n))^2/((2*n)!)^2*3^(4*n-2),n=2..infinity)),18);

11.5167959003610174

In addition, if taking into Eq. (17), then

(25)

>evalf(int((sin(sqrt(5)*sin(x))*cos(sqrt(5)*sin(x)))^2/((sinh(sqrt(5)*cos(x)))^2+(sin(sqrt(5)*sin(x)))^2)^2,x=0..2*Pi),18);

0.531916497721471181

>evalf(Pi*(4/45+sum(2^(4*n)*(bernoulli(2*n))^2/((2*n)!)^2*(sqrt(5))^(4*n-2),n=2..infinity)),18);

0.531916497721471182

On the other hand, let in Eq. (18), then

(26)

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

5.01918539817249445

>evalf(2*Pi*(36/169+169/324)+2*Pi*sum(2^(4*n)*(bernoulli(2*n))^2/((2*n)!)^2*(13/6)^(4*n-2),n=2..infinity),18);

5.01918539817249446

4. Conclusion

In this paper, we use Parseval’s theorem to determine some types of definite integrals. In fact, the applications of this theorem 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 solve these problems by using Maple. 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]	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
[2]	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, pp. 253-260, Vancouver, Canada, 1999.
	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, “Some types of integral problems,” American Journal of Systems and Software, vol. 2, no. 1, pp. 22-26, 2014.
	In article
[7]	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
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	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
[15]	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
[16]	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
[17]	C. -H. Yu, “Application of Maple on evaluation of definite integrals,” Applied Mechanics and Materials, vols. 479-480, pp. 823-827, 2013.
	In article		CrossRef
[18]	C. -H. Yu, “A study of some integral problems using Maple,” Mathematics and Statistics, vol. 2, no. 1, pp. 1-5, 2014.
	In article
[19]	C. -H. Yu, “Application of Maple on the integral problems,” Applied Mechanics and Materials, vols. 479-480, pp. 849-854, 2013.
	In article		CrossRef
[20]	C.-H. Yu, “Application of Maple on the integral problem of some type of rational functions,” Proceedings of the Annual Meeting and Academic Conference for Association of IE, D357-D362, 2012.
	In article
[21]	C. -H. Yu, “Using Maple to study the integrals of trigonometric functions,” Proceedings of the 6th IEEE/International Conference on Advanced Infocomm Technology, no. 00294, 2013.
	In article
[22]	C.-H. Yu, “Application of Maple on some type of integral problem,” Proceedings of the Ubiquitous-Home Conference 2012, pp. 206-210, 2012.
	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, Springer, vol. 1, pp. 603-610, 2013.
	In article
[24]	C.-H. Yu, “Application of Maple on some integral problems,” Proceedings of the International Conference on Safety & Security Management and Engineering Technology 2012, pp. 290-294, 2012.
	In article
[25]	C.-H. Yu, “Application of Maple on evaluating the closed forms of two types of integrals,” Proceedings of the 17th Mobile Computing Workshop, ID 16, 2012.
	In article
[26]	C.-H. Yu, “Application of Maple: taking two special integral problems as examples,” Proceedings of the 8th International Conference on Knowledge Community, pp. 803-811, 2012.
	In article
[27]	R. V. Churchill and J. W. Brown, Complex variables and applications, 4th ed., McGraw-Hill, New York, p 65, 1984.
	In article
[28]	Hyperbolic functions, online available from https://en.wikipedia.org/wiki/Hyperbolic_function
	In article
[29]	D. V. Widder, Advanced calculus, 2nd ed., Prentice-Hall, New Jersey, p 428, 1961.
	In article