06 IJSRST Volume Issue Pit ISSN: 395-60 Olie ISSN: 395-60X 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
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. 0..5 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 (www.ijsst.com) 6
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); 0.0065975737035 >evalf(sum((*)!/(^*(*+)*(*+0)*!*!)*((si (Pi/))^(*+0)-(si(-Pi/3))^(*+0))=0..ifiity) 8); 0.0065975737036 O the othe had usig Eq. (8) yields / 3 si 0 d / 6 3 6 ( )!. 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); 0.078836069306 >evalf(-pi/*(((*pi/3))^-((pi/6))^)+sum(( *)!/(^*(*+)*(*+)*!*!)*(((*Pi/3))^(* +)-((Pi/6))^(*+))=0..ifiity)8); 0.078836069305 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 (www.ijsst.com) 65
Usig Maple to veify the coectess of Eq. (5) as follows: >evalf(it(theta*((theta))^*(ta(theta))^6theta=- Pi/6..Pi/8)); 0.333753607807389 >evalf(sum((-)^/((*+)*(*+8))*((ta(pi/8))^( *+8)-(ta(-Pi/6))^(*+8))=0..ifiity)); 0.0033375360780738 I additio by Eq. (0) we obtai 5 / 9 d / 3 5 5 5 0 9 3 6 6 ( ) 5. 0 ( )( 6) 9 3 (6) >evalf(it(theta*((theta))^*((theta))^theta=pi/3..5*pi/9)8); 0.0899686303799 >evalf(-pi/0*(((5*pi/9))^5-((pi/3))^5)+sum((- )^/((*+)*(*+6))*(((5*Pi/9))^(*+6)-(( Pi/3))^(*+6))=0..ifiity)8); 0.08996863037990 Eample 3 Usig Eq. () yields / 6 ta d /9 6 6 9 5 5 ( )!. 0 ( )( 5)!! 9 (7) We employ Maple to veify the coectess of Eq. (7). >evalf(it(theta*(ta(theta))*((theta))^6theta=pi/9.. Pi/)); 0.7788903035789359 >evalf(pi/*(((pi/))^6-((pi/9))^6)-sum((*)!/( ^*(*+)*(-*+5)*!*!)*(((Pi/))^(-*+5)- ((Pi/9))^(-*+5))=0..ifiity)); 0.7788903035789357 O the othe had by Eq. () we have 3 /8 d /8 ()! 0 ( )( )!! (8) 3. 8 8 Usig Maple to veify Eq. (8) as follows: >evalf(it(theta*(theta)*((theta))^theta=pi/8..3 *Pi/8)8); 36.865838779 >evalf(-sum((*)!/(^*(*+)*(-*+)*!*!)*(( (3*Pi/8))^(-*+)-((Pi/8))^(-*+))=0.. ifiity)8); 36.865838779 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. 999. 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 53-60. [] Nyblom M. A. 007. O the evaluatio of a defiite itegal ivolvig ested squae oot fuctios Rocky Moutai Joual of Mathematics 37() 30-30. [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 () 60-6. [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 (www.ijsst.com) 66
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