Using Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu

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1 Available at Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of Science and Technology, Tainan City, Taiwan chiihuei@mailnjuedutw Abstact In this aticle, we use Maple fo the auxiliay tool to study fou types of impope integals The closed foms of these impope integals can be obtained by using Laplace tansfom On the othe hand, some examples ae poposed to demonstate the calculations, and we veify thei answes using Maple integal poblems, fo example, change of vaiables method, integation by pats method, patial factions method, tigonometic substitution method, etc This pape studies the following fou types of impope integals which ae not easy to obtain thei answes using the methods mentioned above Key Wods: impope integals; closed foms; Laplace tansfom; Maple 1 Intoduction The compute algeba system (CAS) has been widely employed in mathematical and scientific studies The apid computations and the visually appealing gaphical inteface of the pogam ende ceative eseach possible Maple possesses significance among mathematical calculation systems and can be consideed a leading tool in the CAS field The supeioity of Maple lies in its simple instuctions and ease of use, which enable beginnes to lean the opeating techniques in a shot peiod In addition, though the numeical and symbolic computations pefomed by Maple, the logic of thining can be conveted into a seies of instuctions The computation esults of Maple can be used to modify ou pevious thining diections, theeby foming diect and constuctive feedbac that can aid in impoving undestanding of poblems and cultivating eseach inteests In calculus and engineeing mathematics couses, thee ae many methods to solve the t n e xt cos( yt), (1) t n e xt sin( yt), (2) n ( cos t e cos[( sin ], (3) n ( cos t e sin[( sin ], (4) whee x, y,, ae eal numbes, n is a nonnegative intege, x, and cos The closed foms of these fou types of impope integals can be obtained by using Laplace tansfom, these ae the majo esults in this pape: Theoems 1 and 2 Adams et al [1], Nyblom [2], and Oste [3] povided some methods to solve some integal poblems Yu [4-24], Yu and Chen [25], and Yu and Sheu [26-28] used complex powe seies, complex integal fomulas, integation tem by tem, diffeentiation with espect to a paamete, Paseval s theoem, and aea mean value theoem to solve some types of integals In this aticle, we popose some impope integals to do calculation pactically On the othe hand, Maple is used to calculate the appoximations of these impope integals and thei closed foms fo veifying ou answes Available online: P a g e 211

2 Available at Volume 3 Issue 4 Febuay Peliminaies and Results Some notations and fomulas used in this pape ae intoduced below 21 Notations: 211 Let z a ib be a complex numbe, whee i 1, and a, b ae eal numbes a, the eal pat of z, is denoted as Re( z ) ; b, the imaginay pat of z, is denoted as Im(z ) 212 ( ) s( s 1) ( s n 1), whes s is s n a eal numbe, and n is a positive intege; ( s ) 1 22 Fomulas: 221 Eule s fomula: e i cos isin, whee is any eal numbe 222 DeMoive s fomula: n (cos isin ) cosn isin n, whee n is an intege, and is a eal numbe 223 Laplace tansfom ([29, p67]): Suppose that n is a non-negative intege, and s is a complex numbe with Re( s ), then n st n! t e (5) n1 s In the following, we detemine the closed foms of the impope integals (1) and (2) Theoem 1 Suppose that x, y ae eal numbes, n is a non-negative intege, and x, then n xt t e cos( yt) n1 ( 1) ( n 1)! n1 n cos x y, 2 2 n1 ( x y ) (6) and n xt t e sin( yt) n1 ( 1) ( n 1) 1! n n sin x y 2 2 n1 ( x y ) Poof Using Eq (5) yields Thus, (7) n ( xiy) t n! t e (8) n1 ( x iy) n xt t e cos( yt) n! Re n1 ( x iy) (by Eule s fomula) n1 n!re[( x iy) ] 2 2 n1 ( x y ) n1 ( n 1) n 1 n!re x ( iy)! 2 2 n1 ( x y ) n1 ( 1) ( n 1)! n1 n cos x y 2 2 n1 ( x y ) Similaly, by Eq (8) we have n xt t e sin( yt) n! Im n1 ( x iy) n1 ( 1) ( n 1) 1! n n sin x y 2 2 n1 ( x y ) qed The closed foms of the impope integals (3) and (4) can be obtained below Available online: P a g e 212

3 Available at Volume 3 Issue 4 Febuay 216 Theoem 2 If, ae eal numbes, n is a non-negative intege, and cos, then and n ( cos t e cos[( sin ] n!cos[( n 1) ], n1 n ( cos t e sin[( sin ] n!sin[( n 1) ] n1 Poof Using Eq (5) yields Hence, n e i t n! t e i n1 ( e ) n ( cos t e cos[( sin ] n! Re i n1 ( e ) (by Eule s fomula) n! Re n1 i( n1) e (by DeMoive s fomula) n!cos[( n 1) ] n1 Also, using Eq (11) yields n ( cos t e sin[( sin ] n! Im i n1 ( e ) n!sin[( n 1) ] n1 3 Example (9) (1) (11) qed Fo the fou types of impope integals in this study, we will popose some examples and use Theoems 1 and 2 to obtain thei closed foms In addition, Maple is used to calculate the appoximations of these impope integals and thei closed foms fo veifying ou answes Example 1 By Eq (6), we have 3 2t t e cos(5t ) 4 ( 1) (4) 6 cos! (12) The coectness of Eq (12) can be veified by using Maple >evalf(int(t^3*exp(-2*t)*cos(5*t),t= infinity),18); >evalf(246/77281,18); On the othe hand, using Eq (7) yields 6 4t t e sin(7t ) 7 ( 1) (7) 6! sin! (13) We also use Maple to veify the coectness of Eq (13) Available online: P a g e 213

4 Available at Volume 3 Issue 4 Febuay 216 >evalf(int(t^6*exp(-4*t)*sin(7*t),t= infinity),18); >evalf( / ,18); Example 2 We can detemine the following impope integal using Eq (9), 4 [2 cos( /8)] t t e cos[2 sin( /8] 3cos(5 / 8) (14) 4 The coectness of Eq (14) can be veified by using Maple >evalf(int(t^4*exp(-2*cos(pi/8)*t)*cos(2* sin(pi/8)*t),t=infinity),18); >evalf(3*cos(5*pi/8)/4,18); In addition, using Eq(1) yields 7 [3cos( /5)] t t e sin[3sin( / 5] 56sin(8 / 5) (15) 729 >evalf(int(t^7*exp(-3*cos(pi/5)*t)*sin(3* sin(pi/5)*t),t=infinity),18); >evalf(56*sin(8*pi/5)/729,18); 4 Conclusion In this pape, we use Laplace tansfom to solve some impope integals In fact, the applications of Laplace tansfom ae extensive, and can be used to easily solve many difficult poblems; we endeavo to conduct futhe studies on elated applications In addition, Maple also plays a vital assistive ole in poblem-solving In the futue, we will extend the eseach topics to othe calculus and engineeing mathematics poblems and solve these poblems using Maple Refeences: [1] A A Adams, H Gottliebsen, S A Linton, and U Matin, 1999, Automated theoem poving in suppot of compute algeba: symbolic definite integation as a case study, Poceedings of the 1999 Intenational Symposium on Symbolic and Algebaic Computation, Canada, pp [2] M A Nyblom, 27, On the evaluation of a definite integal involving nested squae oot functions, Rocy Mountain Jounal of Mathematics, Vol 37, No 4, pp [3] C Oste, 1991, Limit of a definite integal, SIAM Review, Vol 33, No 1, pp [4] C -H Yu, 214, Solving some definite integals using Paseval s theoem, Ameican Jounal of Numeical Analysis, Vol 2, No 2, pp 6-64 [5] C -H Yu, 214, Some types of integal poblems, Ameican Jounal of Systems and Softwae, Vol 2, No 1, pp Available online: P a g e 214

5 Available at Volume 3 Issue 4 Febuay 216 [6] C -H Yu, 213, Using Maple to study the double integal poblems, Applied and Computational Mathematics, Vol 2, No 2, pp [7] C -H Yu, 213, A study on double integals, Intenational Jounal of Reseach in Infomation Technology, Vol 1, Issue 8, pp [8] C -H Yu, 214, Application of Paseval s theoem on evaluating some definite integals, Tuish Jounal of Analysis and Numbe Theoy, Vol 2, No 1, pp 1-5 [9] C -H Yu, 214, Evaluation of two types of integals using Maple, Univesal Jounal of Applied Science, Vol 2, No 2, pp [1] C -H Yu, 214, Studying thee types of integals with Maple, Ameican Jounal of Computing Reseach Repositoy, Vol 2, No 1, pp [11] C -H Yu, 214, The application of Paseval s theoem to integal poblems, Applied Mathematics and Physics, Vol 2, No 1, pp 4-9 [12] C -H Yu, 214, A study of some integal poblems using Maple, Mathematics and Statistics, Vol 2, No 1, pp 1-5 [13] C -H Yu, 214, Solving some definite integals by using Maple, Wold Jounal of Compute Application and Technology, Vol 2, No 3, pp [14] C -H Yu, 213, Using Maple to study two types of integals, Intenational Jounal of Reseach in Compute Applications and Robotics, Vol 1, Issue 4, pp [15] C -H Yu, 213, Solving some integals with Maple, Intenational Jounal of Reseach in Aeonautical and Mechanical Engineeing, Vol 1, Issue 3, pp [16] C -H Yu, 213, A study on integal poblems by using Maple, Intenational Jounal of Advanced Reseach in Compute Science and Softwae Engineeing, Vol 3, Issue 7, pp [17] C -H Yu, 213, Evaluating some integals with Maple, Intenational Jounal of Compute Science and Mobile Computing, Vol 2, Issue 7, pp [18] C -H Yu, 213, Application of Maple on evaluation of definite integals, Applied Mechanics and Mateials, Vols , pp [19] C -H Yu, 213, Application of Maple on the integal poblems, Applied Mechanics and Mateials, Vols , pp [2] C -H Yu, 213, Using Maple to study multiple impope integals, Intenational Jounal of Reseach in Infomation Technology, Vol 1, Issue 8, pp 1-14 Available online: P a g e 215

6 Available at Volume 3 Issue 4 Febuay 216 [21] C -H Yu, 213, Using Maple to study the integals of tigonometic functions, Poceedings of the 6th IEEE/Intenational Confeence on Advanced Infocomm Technology, Taiwan, No 294 [22] C -H Yu, 214, Evaluating some types of definite integals, Ameican Jounal of Softwae Engineeing, Vol 2, Issue 1, pp Jounal of Systems and Softwae, Vol 2, No 4, pp [29] D Zwillinge, 23, CRC Standad Mathematical Tables and Fomulae, 31 st ed, Floida: CRC Pess [23] C -H Yu, 216, A study of an integal elated to the logaithmic function with Maple, Intenational Jounal of Reseach, Vol 3, Issue 1, pp [24] C -H Yu, 216, Solving eal Integals using complex integals, Intenational Jounal of Reseach, Vol 3, Issue 4, pp 95-1 [25] C -H Yu and B -H Chen, 214, Solving some types of integals using Maple, Univesal Jounal of Computational Mathematics, Vol 2, No 3, pp [26] C -H Yu and S -D Sheu, 214, Using aea mean value theoem to solve some double integals, Tuish Jounal of Analysis and Numbe Theoy, Vol 2, No 3, pp [27] C -H Yu and S -D Sheu, 214, Infinite seies foms of double integals, Intenational Jounal of Data Envelopment Analysis and *Opeations Reseach*, Vol 1, No 2, pp 16-2 [28] C -H Yu and S -D Sheu, 214, Evaluation of tiple integals, Ameican Available online: P a g e 216