Generalization of Two Types of Improper Integrals

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1 World Joural of Computer Applicatio ad Techology 2(5): -03, 204 DOI: 0.38/wjcat Geeralizatio of Two Types of Improper Itegrals Chii-Huei Yu,*, Big-Huei Che 2 Departmet of Maagemet ad Iformatio, Na Jeo Uiversity of Sciece ad Techology, Taia City, 73746, Taiwa 2 Departmet of Electrical Egieerig, Na Jeo Uiversity of Sciece ad Techology, Taia City, 73746, Taiwa *Correspodig Author: chiihuei@mail.ju.edu.tw Copyright 204 Horizo Research Publishig All rights reserved. Abstract This article uses the mathematical software Maple for the auxiliary tool to study two types of improper itegrals. We ca obtai the ifiite series forms of these two types of itegrals by usig geometric series, differetiatio term by term, ad differetiatio with respect to a parameter. O the other had, we provide some examples to do calculatio practically. The research methods adopted i this study ivolved fidig solutios through maual calculatios ad verifyig these solutios by usig Maple. This type of research method ot oly allows the discovery of calculatio errors, but also helps modify the origial directios of thikig from maual ad Maple calculatios. For this reaso, Maple provides isights ad guidace regardig problem-solvig methods. Keywords Improper Itegrals, Ifiite Series Forms, Geometric Series, Differetiatio Term by Term, Differetiatio with Respect to a Parameter, Maple. Itroductio The computer algebra system (CAS) has bee widely employed i mathematical ad scietific studies. The rapid computatios ad the visually appealig graphical iterface of the program reder creative research possible. Maple possesses sigificace amog mathematical calculatio systems ad ca be cosidered a leadig tool i the CAS field. The superiority of Maple lies i its simple istructios ad ease of use, which eable begiers to lear the operatig techiques i a short period. I additio, through the umerical ad symbolic computatios performed by Maple, the logic of thikig ca be coverted ito a series of istructios. The computatio results of Maple ca be used to modify our previous thikig directios, thereby formig direct ad costructive feedback that ca aid i improvig uderstadig of problems ad cultivatig research iterests. Iquirig through a olie support system provided by Maple or browsig the Maple website ( ca facilitate further uderstadig of Maple ad might provide uexpected isights. As for the istructios ad operatios of Maple, we ca refer to [-7]. I [8], there are the followig formulas of two types of improper itegrals si a tah () 0 sihbx cos a sech (2) where a, b are real umbers, ad a 0, b > 0. I this study, we geeralize the above improper itegrals () ad (2) to the followig two types of improper itegrals x si sihbx x cos + 2 (4) where a, b are real umbers, a 0, b > 0, ad is ay o-egative iteger. We ca obtai the ifiite series forms of these two types of improper itegrals by usig geometric series, differetiatio term by term, ad differetiatio with respect to a parameter; these are the major results of this study (i.e., Theorems ad 2). The study of related itegral problems ca refer to [-2]. O the other had, we propose four examples to do calculatio practically. The research methods adopted i this study ivolved fidig solutios through maual calculatios ad verifyig these solutios by usig Maple. This type of research method ot oly allows the discovery of calculatio errors, but also helps modify the origial directios of thikig from maual ad Maple calculatios. Therefore, Maple provides isights ad guidace regardig problem-solvig methods. 2. Mai Results Firstly, we itroduce three importat theorems used i this study. 2.. Geometric Series (3)

2 00 Geeralizatio of Two Types of Improper Itegrals where u is a real umber, ad u < Differetiatio Term by Term ([30]) For all o-egative itegers k, if the fuctios g k : ( a, R satisfy the followig three coditios:(i) there exists a poit x0 ( a, such that g k ( x 0 ) is coverget, (ii) all fuctios g k ( are differetiable o ope iterval ( a,, (iii) d g k ( x ) is uiformly coverget o ( a,. The g k ( x ) is uiformly coverget ad differetiable o ( a,. Moreover, its derivative d g ( ). k x d g k ( x ) 2.3. Differetiatio with Respect to A Parameter ([3]) Suppose c,d,λ, β are real umbers ad the fuctio f ( a, is defied o [ c, d ] [ λ, β ]. If f ( a, ad its f partial derivative ( a, are cotiuous fuctios o a [ c, d ] [ λ, β ]. The F(a) β f ( a, is differetiable λ d o the ope iterval ( c, d ), ad its derivative F(a) da β f ( a, for all a ( c, d ). λ a The followig is the first result i this study, we determie the ifiite series form of the improper itegral (3) Theorem Suppose a, b are real umbers, a 0, b > 0, ad is ay o-egative iteger. Case (A): If a > 0. The the improper itegral x si sihbx + ( ) + k ak ( ) k exp b Case (B): If a < 0 + u. The ( ) k u k k 0, (5) x si sihbx + k ak ( ) k exp + b Proof Case (A): If a > 0. Because si 0 sihbx a tah (By ()) a a a a + + a b a b 2 + a + b (6) + k ak 2 ( ) b (Usig geometric series) (7) + k ak ( ) b By differetiatio term by term ad differetiatio with respect to a parameter, differetiatig -times with respect to a o both sides of (8), we obtai x si sihbx + ( ) + k ak ( ) k exp b Case (B): If a < 0. Because si 0 sihbx a tah a tah (8)

3 World Joural of Computer Applicatio ad Techology 2(5): -03, k ak 2 ( ) b k ak ( ) b (By (7)) Also, by differetiatio term by term ad differetiatio with respect to a parameter, differetiatig -times with respect to a o both sides of (), we obtai x si sihbx + k ak ( ) k exp + b q.e.d. The followig is the secod major result i this paper, we fid the ifiite series form of the improper itegral (4) Theorem 2 Let the assumptios be the same as Theorem. Case (A): If a > 0. The the improper itegral x cos ( ) k a + ) ( ) + ) exp + Case (B): If a < 0. The x cos k a + ) ( ) + ) exp Proof Case (A): If a > 0. Because cos a sech (By (2)) b a + a b a + a b a k ak exp ( ) b k 0 b geometric series) (By () (0) () + k a ) ( ) (2) Usig differetiatio term by term ad differetiatio with respect to a parameter, differetiatig -times with respect to a o both sides of (2), we have x cos ( ) k a + ) ( ) + ) exp + Case (B): If a < 0. Because cos a sech a sech + k a ) ( ) (Usig (2)) (3) Also, by differetiatio term by term ad differetiatio with respect to a parameter, differetiatig -times with respect to a o both sides of (3), we obtai x cos k a + ) ( ) + ) exp + 3. Examples q.e.d. I the followig, for the two types of improper itegrals i this study, we provide four examples ad use Theorems, 2 to determie their ifiite series forms. O the other had, we employ Maple to calculate the approximatios of these improper itegrals ad their solutios for verifyig our aswers. 3.. Example I Theorem, let a 6, b 2, 8. By Case (A) of Theorem, we obtai the improper itegral 0 8 x si 6x k 8 ( ) k exp( 3k ) sih 2x 2 k 0 Next, we use Maple to verify the correctess of (4). >evalf(it(x^8*si(6*/sih(2*,x0..ifiity),4); >evalf(pi^/2^*sum((-)^k*k^8*exp(-3*k*pi),k0. (4)

4 02 Geeralizatio of Two Types of Improper Itegrals ifiity),4); *k+3)*Pi/8),k0..ifiity),8); Example 2 I Theorem, takig a 4, b 5,. Usig Case (B) of Theorem, we ca determie the improper itegral 0 x cos4x sih5x 0 k 4k ( ) k 0 5 k 0 5 We also use Maple to verify the correctess of (5). >evalf(it(x^*cos(4*/sih(5*,x0..ifiity),4); (5) >evalf(-pi^0/5^0*sum((-)^k*k^*exp(-4*k*pi/5),k0..i fiity),4); Coclusios As metioed, the geometric series, the differetiatio term by term, ad the differetiatio with respect to a parameter play sigificat roles i the theoretical ifereces of this study. I fact, the applicatios of these theorems are extesive, ad ca be used to easily solve may difficult problems; we edeavor to coduct further studies o related applicatios. O the other had, Maple also plays a vital assistive role i problem-solvig. I the future, we will exted the research topic to other calculus ad egieerig mathematics problems ad solve these problems by usig Maple. These results will be used as teachig materials for Maple o educatio ad research to ehace the cootatios of calculus ad egieerig mathematics Example 3 I Theorem 2, let a 7, b 4, 2. By Case (A) of Theorem 2, we ca evaluate the improper itegral 0 2 x cos7x cosh4x 3 k 2 (4k + 7) ( ) + ) k 0 8 Usig Maple to verify the correctess of (6) as follows: >evalf(it(x^2*cos(7*/cosh(4*,x0..ifiity),4); >evalf(pi^3/(2^2*4^3)*sum((-)^k*(2*k+)^2*exp(-( 4*k+7)*Pi/8),k0..ifiity),4); 3.4. Example (6) I Theorem 2, let a 6, b 8, 5. Usig Case (B) of Theorem 2, we obtai the improper itegral 0 5 x si 6x cosh8x 6 k 5 (6k + 3) ( ) + ) k 0 8 We also use Maple to verify the correctess of (7). >evalf(it(x^5*si(6*/cosh(8*,x0..ifiity),8); (7) >evalf(-pi^6/(2^5*8^6)*sum((-)^k*(2*k+)^5*exp(-( REFERENCES [] M. L. Abell ad J. P. Braselto, Maple by Example, 3rd ed., New York: Elsevier Academic Press, [2] J. S. Robertso, Egieerig Mathematics with Maple, New York: McGraw-Hill, 6. [3] F. Garva, The Maple Book, Lodo: Chapma & Hall/CRC, 200. [4] R. J. Stroeker ad J. F. Kaashoek, Discoverig Mathematics with Maple : A Iteractive Exploratio for Mathematicias, Egieers ad Ecoometricias, Basel: Birkhauser Verlag,. [5] C. T. J. Dodso ad E. A. Gozalez, Experimets i Mathematics Usig Maple, New York: Spriger-Verlag, 5. [6] D. Richards, Advaced Mathematical Methods with Maple, New York: Cambridge Uiversity Press, [7] C. Tocci ad S. G. Adams, Applied Maple for Egieers ad Scietists, Bosto: Artech House, 6. [8] D. Zwilliger (Ed.), CRC Stadard Mathematical Tables ad Formulae, 3st ed., Boca Rato: Chapma & Hall/CRC, p455, [] A. A. Adams, H. Gottliebse, S. A. Lito, ad U. Marti, Automated theorem provig i support of computer algebra: symbolic defiite itegratio as a case study, Proceedigs of Iteratioal Symposium o Symbolic ad Algebraic Computatio, pp , Vacouver, Caada,. [0] M. A. Nyblom, O the evaluatio of a defiite itegral ivolvig ested square root fuctios, Rocky Moutai Joural of Mathematics, Vol. 37, No. 4, pp , [] C. Oster, Limit of a defiite itegral, SIAM Review, Vol. 33, No., pp. 5-6,.

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