Integral Problems of Trigonometric Functions

Transcription

1 06 IJSRST Volume Issue Pit ISSN: Olie ISSN: X Themed Sectio: Sciece ad Techology Itegal Poblems of Tigoometic Fuctios Chii-Huei Yu Depatmet of Ifomatio Techology Na Jeo Uivesity of Sciece ad Techology Taia City Taiwa ABSTRACT The aticle ides si types of itegals elated with the powes of tigoometic fuctios. We ca obtai the ifiite seies epessios of these itegals by usig Taylo seies epasios ad itegatio tem by tem theoem. Moeove we popose some itegals to do calculatio ad evaluate some defiite itegals pactically. O the othe had Maple is used to calculate the appoimatios of these defiite itegals ad thei ifiite seies epessios fo veifyig ou aswes. Keywods: Itegals Tigoometic Fuctios Ifiite Seies Epessios Taylo Seies Epasios Itegatio Tem by Tem Theoem Maple I. INTRODUCTION As ifomatio techology advaces whethe computes ca become compaable with huma bais to pefom abstact tasks such as abstact at simila to the paitigs of Picasso ad musical compositios simila to those of Beethove is a atual questio. Cuetly this appeas uattaiable. I additio whethe computes ca solve abstact ad difficult mathematical poblems ad develop abstact mathematical theoies such as those of mathematicias also appeas ufeasible. Nevetheless i seekig fo alteatives we ca study what assistace mathematical softwae ca povide. This study itoduces how to coduct mathematical eseach usig the mathematical softwae Maple. The mai easos of usig Maple i this study ae its simple istuctios ad ease of use which eable begies to lea the opeatig techiques i a shot peiod. By employig the poweful computig capabilities of Maple difficult poblems ca be easily solved. Eve whe Maple caot detemie the solutio poblem-solvig hits ca be idetified ad ifeed fom the appoimate values calculated ad solutios to simila poblems as detemied by Maple. Fo this easo Maple ca povide isights ito scietific eseach. I calculus ad egieeig mathematics thee ae may methods to solve the idefiite itegals icludig chage of vaiables method itegatio by pats method patial factios method tigoometic substitutio method etc. This pape ides the followig si types of itegals elated with the powes of tigoometic fuctios which ae ot easy to obtai thei aswes usig the methods metioed above. si d () whee si d ta d d ta d d () (3) () (5) (6) ae eal umbes. The ifiite seies epessios of these itegals ca be obtaied maily usig Taylo seies epasios ad itegatio tem by tem theoem; these ae the majo esults of this aticle (i.e. Theoems -3). Adams et al. [] Nyblom [] ad Oste [3] povided some techiques to solve the itegal poblems. Moeove Yu [-3] Yu ad Che [3] ad Yu ad Sheu [33-35] used comple powe seies method itegatio tem by tem theoem Paseval s theoem aea mea value theoem ad geealized Cauchy itegal fomula to evaluate some types of itegal poblems. I this pape some eamples ae used IJSRST6 Received: 09 Febuay 06 Accepted: 7 Febuay 06 Jauay-Febuay-06 [(): 63-67] 63

2 to demostate the poposed calculatios ad the maual calculatios ae veified usig Maple. II. PRELIMINARIES AND RESULTS Fomulas ad Theoems: The followigs ae the Taylo seies epasios of si ivese tigoometic fuctios:.. Ivese sie fuctio ( )! si whee 0 ( )!!... Ivese ie fuctio ( )! whee 0 ( )!!...3 Ivese taget fuctio ( ) ta whee. 0.. Ivese aget fuctio ( ) whee Ivese at fuctio ( )! 0 ( )!! whee...6 Ivese ecat fuctio ( )! whee 0 ( )!!...7 Itegatio tem by tem theoem ([36 p69]): Suppose that g 0 is a sequece of Lebesgue itegable fuctios defied o I. If I 0 coveget the I g 0 I 0 g. g is I the followig we detemie the ifiite seies epessios of the itegals () ad (). Theoem Suppose that is ot a egative eve itege the si d ae eal umbes ad ( )! si C 0 ( )( )!! (7) whee / / ad si eists. si d ( ) ( )! C 0 ( )( )!! (8) whee 0 ad eists. Poof si d si d (whee si ) ()! d 0 ( )!! (by Fomula..) ()! C 0 ( )( )!! (by itegatio tem by tem theoem) ( )! si C. 0 ( )( )!! O the othe had si d d (whee ) ()! d 0 ( )!! (by Fomula..) ( )! C ( ) 0 ( )( )!! (by itegatio tem by tem theoem) ( )! C. ( ) 0 ( )( )!! q.e.d. Usig the same poof as Theoem we ca easily obtai the ifiite seies epessios of the itegals (3) () (5) ad (6) espectively. Iteatioal Joual of Scietific Reseach i Sciece ad Techology ( 6

3 Theoem If the assumptios ae the same as Theoem the ta d ( ) ta C 0 ( )( ) (9) whee / / ad ta eists. d ( ) C ( ) 0 ( )( ) (0) whee / 3 / ad eists. Theoem 3 If ae eal umbes ad is ot a o-egative eve itege the ta d ( ) ( )! C 0 ( )( )!! () whee 0 / ad eists. d ( )! C 0 ( )( )!! () whee / / 0 ad eists. III. EXAMPLES I the followig fo the si types of itegals i this pape we will popose some eamples ad use Theoems -3 to obtai thei ifiite seies epessios. O the othe had we use Maple to calculate the appoimatios of some defiite itegals ad thei solutios fo veifyig ou aswes. Eample By Eq. (7) we have / si 8 d / 3 ()! 0 ( )( 0)!! 0 0 si si. 3 Net we use Maple to veify the coectess of Eq. (3). (3) > evalf(it(theta*(theta)*(si(theta))^8theta=-pi/3.. Pi/)8); >evalf(sum((*)!/(^*(*+)*(*+0)*!*!)*((si (Pi/))^(*+0)-(si(-Pi/3))^(*+0))=0..ifiity) 8); O the othe had usig Eq. (8) yields / 3 si 0 d / ( )!. 0 ( )( )!! 3 6 () We also use Maple to veify the coectess of Eq. (). >evalf(it(theta*si(theta)*((theta))^0theta=pi/6.. *Pi/3)8); >evalf(-pi/*(((*pi/3))^-((pi/6))^)+sum(( *)!/(^*(*+)*(*+)*!*!)*(((*Pi/3))^(* +)-((Pi/6))^(*+))=0..ifiity)8); Eample It follows fom Eq. (9) that / 8 ta 6 d / 6 ( ) 0 ( )( 8) 8 8 ta ta. 8 6 (5) Iteatioal Joual of Scietific Reseach i Sciece ad Techology ( 65

4 Usig Maple to veify the coectess of Eq. (5) as follows: >evalf(it(theta*((theta))^*(ta(theta))^6theta=- Pi/6..Pi/8)); >evalf(sum((-)^/((*+)*(*+8))*((ta(pi/8))^( *+8)-(ta(-Pi/6))^(*+8))=0..ifiity)); I additio by Eq. (0) we obtai 5 / 9 d / ( ) 5. 0 ( )( 6) 9 3 (6) >evalf(it(theta*((theta))^*((theta))^theta=pi/3..5*pi/9)8); >evalf(-pi/0*(((5*pi/9))^5-((pi/3))^5)+sum((- )^/((*+)*(*+6))*(((5*Pi/9))^(*+6)-(( Pi/3))^(*+6))=0..ifiity)8); Eample 3 Usig Eq. () yields / 6 ta d / ( )!. 0 ( )( 5)!! 9 (7) We employ Maple to veify the coectess of Eq. (7). >evalf(it(theta*(ta(theta))*((theta))^6theta=pi/9.. Pi/)); >evalf(pi/*(((pi/))^6-((pi/9))^6)-sum((*)!/( ^*(*+)*(-*+5)*!*!)*(((Pi/))^(-*+5)- ((Pi/9))^(-*+5))=0..ifiity)); O the othe had by Eq. () we have 3 /8 d /8 ()! 0 ( )( )!! (8) Usig Maple to veify Eq. (8) as follows: >evalf(it(theta*(theta)*((theta))^theta=pi/8..3 *Pi/8)8); >evalf(-sum((*)!/(^*(*+)*(-*+)*!*!)*(( (3*Pi/8))^(-*+)-((Pi/8))^(-*+))=0.. ifiity)8); IV. CONCLUSION I this study we use Taylo seies epasios ad itegatio tem by tem theoem to solve some types of itegals. I fact the applicatios of the two methods ae etesive ad ca be used to easily solve may difficult poblems; we edeavo to coduct futhe studies o elated applicatios. O the othe had Maple also plays a vital assistive ole i poblem-solvig. I the futue we will eted the eseach topics to othe calculus ad egieeig mathematics poblems ad use Maple to veify ou aswes. V. REFERENCES [] Adams A. A. Gottliebse H. Lito S. A. ad Mati U Automated theoem povig i suppot of compute algeba: symbolic defiite itegatio as a case study Poceedigs of the 999 Iteatioal Symposium o Symbolic ad Algebaic Computatio Caada [] Nyblom M. A O the evaluatio of a defiite itegal ivolvig ested squae oot fuctios Rocky Moutai Joual of Mathematics 37() [3] Oste C. 99. Limit of a defiite itegal SIAM Review 33() 5-6. [] Yu C. -H. 0. Solvig some defiite itegals usig Paseval s theoem Ameica Joual of Numeical Aalysis () [5] Yu C. -H. 0. Some types of itegal poblems Ameica Joual of Systems ad Softwae () -6. Iteatioal Joual of Scietific Reseach i Sciece ad Techology ( 66

5 [6] Yu C. -H. 0. Applicatio of Paseval s theoem o evaluatig some defiite itegals Tukish Joual of Aalysis ad Numbe Theoy () -5. [7] Yu C. -H. 0. Evaluatio of two types of itegals usig Maple Uivesal Joual of Applied Sciece () [8] Yu C. -H. 0. Studyig thee types of itegals with Maple Ameica Joual of Computig Reseach Repositoy () 9-. [9] Yu C. -H. 0. The applicatio of Paseval s theoem to itegal poblems Applied Mathematics ad Physics () -9. [0] Yu C. -H. 0. A study of some itegal poblems usig Maple Mathematics ad Statistics () -5. [] Yu C. -H. 0. Solvig some defiite itegals by usig Maple Wold Joual of Compute Applicatio ad Techology (3) [] Yu C. -H. 0. Evaluatig some types of defiite itegals Ameica Joual of Softwae Egieeig () 3-5. [3] Yu C. -H. 03. Usig Maple to study two types of itegals Iteatioal Joual of Reseach i Compute Applicatios ad Robotics () -. [] Yu C. -H. 03. Solvig some itegals with Maple Iteatioal Joual of Reseach i Aeoautical ad Mechaical Egieeig Vol. (3) pp [5] Yu C. -H. 03. A study o itegal poblems by usig Maple Iteatioal Joual of Advaced Reseach i Compute Sciece ad Softwae Egieeig 3(7) -6. [6] Yu C. -H. 03. Evaluatig some itegals with Maple Iteatioal Joual of Compute Sciece ad Mobile Computig (7) [7] Yu C. -H. 03. Applicatio of Maple o evaluatio of defiite itegals Applied Mechaics ad Mateials [8] Yu C. -H. 03. Applicatio of Maple o the itegal poblems Applied Mechaics ad Mateials [9] Yu C. -H. 03. Usig Maple to study multiple impope itegals Iteatioal Joual of Reseach i Ifomatio Techology (8) 0-. [0] Yu C. -H. 03. A study o the multiple impope itegal poblems (i Chiese) Joual of Hsi Sheg [] Yu C. -H. 03. Applicatio of Maple: the evaluatio of double itegal as a eample (i Chiese) Poceedigs of 03 Iteatioal Symposium o Itecultual Commuicatio Taiwa [] Yu C. -H. 03. Usig Maple to study the double itegal poblems Applied ad Computatioal Mathematics () 8-3. [3] Yu C. -H. 03. A study o double itegals Iteatioal Joual of Reseach i Ifomatio Techology (8) -3. [] Yu C. -H. 03. Usig Maple to study the itegals of tigoometic fuctios Poceedigs of the 6th IEEE/Iteatioal Cofeece o Advaced Ifocomm Techology Taiwa No [5] Yu C. -H. 03. A study of the itegals of tigoometic fuctios with Maple Poceedigs of the Istitute of Idustial Egiees Asia Cofeece 03 Taiwa Spige [6] Yu C. -H. 03. Applicatio of Maple o evaluatig the double impope itegals (i Chiese) Poceedigs of the Iovative Educatio ad Leaig Techology Taiwa [7] Yu C. -H. 0. Applicatio of Maple o the itegal poblem of some type of atioal fuctios (i Chiese) Poceedigs of the Aual Meetig ad Academic Cofeece fo Associatio of IE Taiwa D357-D36. [8] Yu C. -H. 0. Applicatio of Maple o some itegal poblems (i Chiese) Poceedigs of the Iteatioal Cofeece o Safety & Secuity Maagemet ad Egieeig Techology 0 Taiwa [9] Yu C. -H. 0. Applicatio of Maple o some type of itegal poblem (i Chiese) Poceedigs of the Ubiquitous-Home Cofeece 0 Taiwa [30] Yu C. -H. 0. Applicatio of Maple o evaluatig the closed foms of two types of itegals (i Chiese) Poceedigs of the 7th Mobile Computig Wokshop Taiwa ID6. [3] Yu C. -H. 0. Applicatio of Maple: takig two special itegal poblems as eamples (i Chiese) Poceedigs of the 8th Iteatioal Cofeece o Kowledge Commuity Taiwa [3] Yu C. -H. ad Che B. -H. 0. Solvig some types of itegals usig Maple Uivesal Joual of Computatioal Mathematics (3) [33] Yu C. -H. ad Sheu S. -D 0. Usig aea mea value theoem to solve some double itegals Tukish Joual of Aalysis ad Numbe Theoy (3) [3] Yu C. -H. ad Sheu S. -D 0. Ifiite seies foms of double itegals Iteatioal Joual of Data Evelopmet Aalysis ad *Opeatios Reseach* () 6-0. [35] Yu C. -H. ad Sheu S. -D 0. Evaluatio of tiple itegals Ameica Joual of Systems ad Softwae () [36] Apostol T. M Mathematical Aalysis d ed. Massachusetts: Addiso-Wesley. Iteatioal Joual of Scietific Reseach i Sciece ad Techology ( 67