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 three types of definite integrals which are not easy to obtain their answers using the methods mentioned above.

(1)

(2)

(3)

where is any real number. We can obtain the infinite series forms of these definite integrals using Parseval’s theorem; this is the major result 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 [4-26]^[4], Yu and B. -H. Chen ^[27], and T. -J. Chen and Yu ^{[28, 29, 30]} used complex power series method, integration term by term theorem, differentiation with respect to a parameter, 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

A notation, a definition and some formulas used in this paper are introduced below.

Let be a complex number, where , and are real numbers. , the real part of , denoted as Re(z), and b, the imaginary part of z, denoted as Im(z).

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

, where is any real number.

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

, where is any complex number.

The following is an important theorem used in this article, which can be found in [^[31], p 428].

If f(x) is a continuous function defined on [0, 2π] and f(0)=f(2π). Suppose that the Fourier series expansion of is , then .

Before the major result can be derived, a lemma is needed.

Suppose that is any real number. The Fourier series expansions of the following trigonometric functions are:

(4)

(5)

Proof

Similarly,

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

Suppose that is any real number, the definite integrals:

(6)

(7)

(8)

Proof Using Parseval’s theorem, (4) and (5), we obtain (6) and (7). Moreover, by the summation of (6) and (7), we obtain (8).

3. Examples

In the following, for the three types of definite integrals in this study, we provide some definite integrals and use Theorem A to determine their infinite series forms. On the other hand, we use Maple to calculate the approximations of these definite integrals and their solutions for verifying our answers.

If in (6), we obtain the definite integral:

(9)

Next, we use Maple to verify the correctness of (9).

>evalf(int(exp(10*cos(x))*(cos(5*sin(x)))^2,x=0..2*Pi),

18);

8848.97626723379613.

>evalf(2*Pi+Pi*sum(5^(2*k)/(k!)^2,k=1..infinity),18);

8848.97626723379613.

On the other hand, let in (7), we have:

(10)

>evalf(int(exp(2*sqrt(3)*cos(x))*(sin(sqrt(3)*sin(x)))^2,x=0..2*Pi),18);

19.3490582735096348.

>evalf(Pi*sum(sqrt(3)^(2*k)/(k!)^2,k=1..infinity),18);

19.3490582735096347.

Finally, if in (8), then the definite integral:

(11)

>evalf(int(exp(15/2*cos(x)),x=0..2*Pi),18);

1684.90721246624587.

>evalf(2*Pi+2*Pi*sum((15/4)^(2*k)/(k!)^2,k=1..infinity),18);

1684.90721246624588.

4. Conclusion

This article uses Parseval’s theorem to evaluate some 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 using Maple.

References

[1]	C. Oster, “Limit of a definite integral,” SIAM Review, vol. 33, no. 1, pp. 115-116, 1991.
	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]	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
[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, “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, “A study of some integral problems using Maple,” Mathematics and Statistics, vol. 2, no. 1, pp. 1-5, 2014.
	In article
[14]	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
[15]	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
[16]	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
[17]	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
[18]	C. -H. Yu, “Application of Maple on the integral problems,” Applied Mechanics and Materials, vols. 479-480, pp. 849-854, 2013.
	In article		CrossRef
[19]	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
[20]	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
[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.2 06-210, 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]	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
[28]	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
[29]	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
[30]	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
[31]	D. V. Widder, Advanced calculus, 2nd ed., Prentice-Hall, New Jersey, 1961.
	In article