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 triple integrals which are not easy to obtain their answers using the methods mentioned above.

(1)

(2)

where are real numbers, , , are real numbers, , , and are positive integers. We can obtain the infinite series forms of these triple integrals using binomial series and integration term by term theorem; these are the major results of this paper (i.e., Theorems 2.5 and 2.6). 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]}, and T. -J. Chen and Yu ^{[33, 34, 35]} used 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 paper, two examples are used to demonstrate the proposed calculations, and the manual calculations are verified using Maple.

2. Main Results

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

Suppose that is a positive real number, then

2.1.2. , where is a real number, and is a positive integer; .

, where x is any real number.

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

Two important theorems used in this study are introduced below.

, where is a complex number, , and is a real number.

The following theorem can be found in [^[36], p 269]

Suppose that is a sequence of Lebesgue integrable functions defined on I. If is convergent, then .

Before deriving the major results of this study, we need a lemma.

Lemma 2.4. Suppose that is a complex number, , are real numbers, , , and are positive integers. Then the improper integral:

(3)

Proof

(4)

Thus,

Firstly, we determine the infinite series form of the triple integral (1).

Theorem 2.5. Assume that are real numbers, , , are real numbers, , , and are positive integers. Then the triple integral:

(5)

Proof Let in Eq. (3), where are real numbers, and , then:

(6)

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

(7)

Therefore,

(8)

Using the equality of real parts of both sides of Eq. (8) yields:

(9)

Hence, the triple integral:

Next, we determine the infinite series form of the triple integral (2).

Theorem 2.6. If the assumptions are the same as Theorem 2.5, then the triple integral:

(10)

Proof Using the equality of imaginary parts of both sides of Eq. (8) yields:

(11)

Thus, the triple integral:

3. Examples

In the following, for the two types of triple integrals in this study, we provide two examples and use Theorems 2.5 and 2.6 to find their infinite series forms. On the other hand, Maple is used to calculate the approximations of these triple integrals and their solutions for verifying our answers.

Example 3.1 In Eq. (5), if ,, then the triple integral:

(12)

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

>evalf(Tripleint(t^2*(exp(t)*cos(4*theta)+2*r*cos(3*theta)+exp(-t)*r^2*cos(2*theta))/(exp(2*t)+2*r*exp(t)* cos(theta)+r^2)^2,t=0..infinity,r=1/3..1/2,theta=Pi/6..Pi/3),14);

-0.0042223802575385

>evalf(2*sum(product(-2-j,j=0..(k-1))*((1/2)^(k+1)-(1/3)^(k+1))*(sin((k+4)*Pi/3)-sin((k+4)*Pi/6))/ (k!*(k+1)*(k+4)*(k+3)^3),k=0..infinity),14);

-0.0042223802575385

Example 3.2. In Eq. (10), if ,, then the triple integral:

(13)

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

>evalf(Tripleint(t*(exp(t)*sin(3*theta)+3*r*sin(2*theta)+3*exp(-t)*r^2*sin(theta))/(exp(2*t)+2*r*exp(t)* cos(theta)+r^2)^3,t=0..infinity,r=1/5..1/3,theta=Pi/4..2*Pi/3),14);

-0.0026286382283423

>evalf(-sum(product(-3-j,j=0..(k-1))*((1/3)^(k+1)-(1/5)^(k+1))*(cos((2*k+6)*Pi/3)-cos((k+3)*Pi/4))/ (k!*(k+1)*(k+3)*(k+5)^2),k=0..infinity),14);

-0.0026286282283423

4. Conclusion

In this paper, we use binomial series and integration term by term theorem to solve two types of triple integrals. In fact, the applications of the two methods 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.

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